Chromatic Polynomials of Fn&times;P2 &nbsp;Graphs: Algebraic Analysis and Scheduling Applications

Sarah M. Talab; Nabeel E. Arif

doi:10.12688/f1000research.176896.2

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Research Article

Revised

Chromatic Polynomials of Fn×P2 Graphs: Algebraic Analysis and Scheduling Applications

[version 2; peer review: 2 approved with reservations]

Sarah M. Talab ¹, Nabeel E. Arif¹

PUBLISHED 12 Jun 2026

Author details Author details

¹ Mathematics, Tikrit University, Tikrit, Saladin Governorate, 34001, Iraq

Sarah M. Talab
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Resources, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

Nabeel E. Arif
Roles: Supervision

OPEN PEER REVIEW

REVIEWER STATUS

This article is included in the Fallujah Multidisciplinary Science and Innovation gateway.

Abstract

Background

Chromatic polynomials are fundamental algebraic invariants in graph theory, linking structural properties of graphs with algebraic and enumerative information. While extensive results exist for paths, cycles, and several classical graph products, the Cartesian product F n × P 2 , where F n is the friendship graph, and P 2 is the path on two vertices, has received limited direct attention despite its layered triangular structure.

Methods

We use a recursive block decomposition of F n × P 2 . After fixing the colors of the two central vertices, we derive the local extension count for the newly attached block through a structured combinatorial case analysis. This yields the transition polynomial ψ ( k ) , which governs both the recurrence relation and the closed-form expression for the chromatic polynomial.

Results

We establish the recurrence relation P n ( k ) = ψ ( k ) P n − 1 ( k ) and the closed-form expression P n ( k ) = [ ψ ( k ) ] n − 2 P 2 ( k ) , where ψ ( k ) = k 4 − 8 k 3 + 26 k 2 − 41 k + 26 . We prove that the chromatic number is χ ( F n × P 2 ) = 3 . The transition polynomial ψ ( k ) has exactly two real roots, both lying in the interval [ 2 , 3 ] . The asymptotic behavior is given by lim n → ∞ | P n ( k ) | 1 n = | ψ ( k ) | for fixed k with P 2 ( k ) ≠ 0 . Numerical computations for selected integer values of k confirm the recurrence and closed-form formula.

Conclusions

This work provides an algebraic characterization of the chromatic polynomial of F n × P 2 through a recursive block structure and a single transition polynomial. The results also support an illustrative two-period scheduling interpretation, where the chromatic polynomial counts feasible room assignments under explicitly stated conflict constraints.

Keywords

Graph coloring, Chromatic polynomial, Cartesian product, Friendship graph, Combinatorial mathematics, Recurrence relation, Closed-form expression, Scheduling.

Corresponding author: Sarah M. Talab

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2026 M. Talab S and E. Arif N. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: M. Talab S and E. Arif N. Chromatic Polynomials of Fn×P2 Graphs: Algebraic Analysis and Scheduling Applications [version 2; peer review: 2 approved with reservations]. F1000Research 2026, 15:351 (https://doi.org/10.12688/f1000research.176896.2) First published: 04 Mar 2026, 15:351 (https://doi.org/10.12688/f1000research.176896.1) Latest published: 12 Jun 2026, 15:351 (https://doi.org/10.12688/f1000research.176896.2)

Revised Amendments from Version 1

This revised version addresses the reviewers’ comments by strengthening the derivation of the transition polynomial ψ(k), adding a formal conditional-independence lemma, and replacing the earlier brief argument with a structured combinatorial case analysis. The computational complexity statement has been clarified as applying only to fixed-k numerical evaluation, and the asymptotic-growth result has been refined to distinguish the general absolute-value formulation from the positive integer coloring range. The scheduling section has been revised as an illustrative two-period model with explicit vertex-to-session and edge-to-conflict mappings. The interpretation of ψ(k) has also been corrected: it is now presented as the fixed-room team-growth factor, while the effect of increasing the number of rooms is measured separately by ratios of chromatic polynomial values. We also revised the Discussion, corrected notation and formatting, and improved the presentation of tables and formulas.

See the authors' detailed response to the review by Wafiq Hibi
See the authors' detailed response to the review by Siti Amiroch

1. Introduction

Chromatic polynomials are fundamental objects in algebraic graph theory. Introduced by Birkhoff (1912) in connection with the four-color problem, they were further developed through Whitney’s (1932) deletion–contraction recurrence and Read’s (1968) systematic studies, culminating in the comprehensive treatment of Tutte and Read (1988). For a graph $G$ , the chromatic polynomial $P (G, k)$ counts the number of proper $k$ -colorings of $G$ , thereby connecting combinatorial structure with algebraic and enumerative properties.

Beyond their theoretical importance, chromatic polynomials and graph coloring methods arise in several applied settings, including task scheduling,¹ data analysis,² network design,³ theoretical chemistry,⁴ and statistical physics.⁵ However, computing the chromatic polynomial is #P-hard in general. This difficulty is especially relevant for graph Cartesian products, where the chromatic polynomial of a product graph is not determined by a simple formula involving only the chromatic polynomials of its factors. Consequently, explicit formulas are usually obtained only for structured graph families.

Cartesian products of graphs provide an important setting for studying layered and composite structures. Their structural and coloring properties have been investigated extensively.⁶^,⁷ Recent studies have derived chromatic polynomials for particular families such as triangular snake graphs, $n$ -centipede graphs, layered graphs, and grid-type products using recursive, transfer-matrix, or decomposition-based methods.⁸^–¹²

While many results are known for Cartesian products involving paths and cycles, the graph family $F_{n} \times P_{2}$ , formed from the friendship graph $F_{n}$ and the path $P_{2}$ , has received limited direct attention. This family combines local triangular clusters with a two-layer product structure, making it suitable for recursive chromatic analysis.

This paper provides an analytical framework for $F_{n} \times P_{2}$ by:

1. deriving a recurrence relation and closed-form expression for its chromatic polynomial;
2. establishing its structural properties and chromatic number;
3. analyzing the transition polynomial, its real roots, and the asymptotic growth rate;
4. validating the formulas through numerical computation;
5. presenting an illustrative two-period scheduling interpretation based on the graph-coloring model.

The scheduling interpretation is motivated by established graph-coloring approaches to scheduling and resource allocation 1, 13, and illustrates how the derived chromatic polynomial can be used to quantify feasible room assignments in a two-period resource-allocation setting with explicitly defined conflict constraints. This applied perspective complements the algebraic results by showing how the recurrence and closed-form expression translate into concrete scheduling counts.

2. Preliminaries

Definition 2.1

¹⁴: The friendship graph $F_{n}$ , for $n \geq 2$ , is defined as the union of $n$ copies of $C_{3}$ sharing a common vertex. This common vertex is called the central vertex.

Formally:

V (F_{n}) = {v_{0}} \cup {u_{i}, w_{i}}_{i = 1}^{n}

where $v_{0}$ is the center vertex, and

E (F_{n}) = {(v_{0}, u_{i}), (v_{0}, w_{i}), (u_{i}, w_{i})}_{i = 1}^{n} .

Hence,

| V (F_{n}) | = 2 n + 1,

and

| E (F_{n}) | = 3 n .

Figure 1 (Friendship graphs F_n):

Definition 2.2

⁶: The Cartesian product of two graphs $G$ and $H$ , denoted by $G \times H$ , is the graph whose vertex set is $V (G) \times V (H)$ , with two vertices $(u_{1}, w_{1})$ and $(u_{2}, w_{2})$ adjacent if:

• $u_{1} = u_{2}$ and $w_{1}$ is adjacent to $w_{2}$ in $H$ , or
• $w_{1} = w_{2}$ and $u_{1}$ is adjacent to $u_{2}$ in $G$ .

Throughout this paper, the symbol $\times$ denotes the Cartesian product of graphs.

Definition 2.3

¹⁵: The chromatic polynomial $P (G, k)$ is a polynomial in $k$ that expresses the number of proper vertex $k$ -colorings of $G$ , such that adjacent vertices share distinct colors.

Definition 2.4:

The graph $F_{n} \times P_{2}$ is the Cartesian product of a friendship graph $F_{n}$ with a path $P_{2}$ , forming two parallel layers of $F_{n}$ with corresponding vertices connected by edges.

Figure 2 (Cartesian product F_n × P₂):

Figure 1. Friendship graphs $F_{n}$ for $n = 2, 3, 4, 5 .$

Each graph is formed by $n$ triangles sharing a common central vertex $v_{0}$ , illustrating the recursive structure of the friendship graph family.

Figure 2. Structure of the Cartesian Product $F_{n} \times P_{2}$ for $n = 2, 3, 4, 5 .$

This construction yields two parallel layers of $F_{n}$ , with corresponding vertices connected by vertical edges.

3. Methods

3.1 Analytical framework and structural decomposition

This is a theoretical study in algebraic and combinatorial graph theory, analyzing the chromatic polynomial of the graph family $G_{n} = F_{n} \times P_{2} .$ The core of our approach is a structural decomposition that reveals a recursive block construction. The graph $G_{n}$ consists of two copies of $F_{n}$ , denoted by layers $A$ and $B$ , with corresponding vertices connected by vertical edges. For $n \geq 3$ , the graph $G_{n}$ is obtained from $G_{n - 1}$ by attaching a new block $B_{n}$ . This block contains the four new peripheral vertices ${u_{n}^{A}, w_{n}^{A}, u_{n}^{B}, w_{n}^{B}}$ together with the edges forming the two new triangles, one in each layer, and the two vertical edges between corresponding peripheral vertices. The block $B_{n}$ meets the previously constructed graph only through the two central vertices $v_{0}^{A}$ and $v_{0}^{B}$ . This localized attachment is the key structural feature that isolates the chromatic contribution of each recursive step.

Lemma 3.1.1

(Conditional independence of the recursive block)

Let $B_{n}$ be the block added when passing from $G_{n - 1}$ to $G_{n}$ in the recursive construction of $G_{n} = F_{n} \times P_{2} .$ The block $B_{n}$ contains the four new peripheral vertices ${u_{n}^{A}, w_{n}^{A}, u_{n}^{B}, w_{n}^{B}} .$ The only vertices of $G_{n - 1}$ adjacent to vertices of $B_{n}$ are the two central vertices $v_{0}^{A}, v_{0}^{B} .$ Consequently, once distinct colors are assigned to $v_{0}^{A}$ and $v_{0}^{B}$ , every coloring constraint involving the new peripheral vertices is local to $B_{n}$ . Hence the number of proper extensions to $B_{n}$ depends only on the number of colors $k$ , and is denoted by

N_{diff} (k) .

Proof: By construction, the $n$ -th triangle of the friendship graph shares only the central vertex $v_{0}$ with the preceding triangles. Therefore, in the Cartesian product $F_{n} \times P_{2}$ , no new peripheral vertex in $B_{n}$ is adjacent to any peripheral vertex of an earlier block.

The only connections between the new block and the previously constructed graph occur through the two central vertices $v_{0}^{A}, v_{0}^{B} .$ Thus, after the colors of $v_{0}^{A}$ and $v_{0}^{B}$ are fixed, the coloring constraints involving the four new vertices are exactly the triangle constraints in the two layers together with the two vertical constraints between corresponding new peripheral vertices. None of these constraints involves a peripheral vertex from any earlier block.

Since $v_{0}^{A}$ and $v_{0}^{B}$ are adjacent, they receive distinct colors in every proper coloring. Moreover, by symmetry of the color palette, the number of valid completions depends only on $k$ , not on the particular labels assigned to $v_{0}^{A}$ and $v_{0}^{B}$ . Therefore, every proper coloring of $G_{n - 1}$ admits the same number of extensions to $B_{n}$ , namely $N_{diff} (k)$ . This proves the claim. ∎

Combinatorial derivation of the transition polynomial $ψ (k)$

3.2.

We now compute $N_{diff} (k)$ , the number of admissible color extensions to the block $B_{n}$ . Since $v_{0}^{A}$ and $v_{0}^{B}$ are adjacent, we may assume by symmetry that

c (v_{0}^{A}) = 1, c (v_{0}^{B}) = 2 .

Put

x = c (u_{n}^{A}), y = c (w_{n}^{A}), z = c (u_{n}^{B}), t = c (w_{n}^{B}) .

The coloring constraints are

x \neq 1, y \neq 1, x \neq y,

z \neq 2, t \neq 2, z \neq t,

and

x \neq z, y \neq t .

Thus $N_{diff} (k)$ is the number of ordered quadruples $(x, y, z, t)$ satisfying these constraints. We count according to whether the ordered pair $(x, y)$ contains the color $2$ .

Case 1. The pair $(x, y)$ contains the color $2$

Since $x \neq y$ , the color $2$ appears in exactly one position. The other color is chosen from the $k - 2$ colors different from $1$ and $2$ . Hence there are

2 (k - 2)

choices for (x, y).

For each such choice, the number of admissible choices for $(z, t)$ is ${(k - 2)}^{2}$ . Indeed, if $x = 2$ and $y = r$ , where $r \notin {1, 2}$ , then $z \neq 2$ , while $t \neq 2, r$ . Before imposing $z \neq t$ , this gives

(k - 1) (k - 2)

choices. The forbidden equality $z = t .$ occurs for exactly $k - 2$ common colors, namely all colors different from $2$ and $r$ Therefore,

(k - 1) (k - 2) - (k - 2) = {(k - 2)}^{2} .

The same count holds when $y = 2$ . Thus, the contribution of this case is $2 {(k - 2)}^{3} .$

Case 2. The pair $(x, y)$ does not contain the color $2$

In this case, x and y are distinct colors chosen from the k−2 colors different from 1 and 2

Hence

(k - 2) (k - 3)

choices for (x, y).

For fixed such $x$ and $y$ , the color $z$ must avoid $2$ and $x$ , while $t$ must avoid $2$ and $y$ . Hence, before imposing $z \neq t$ , there are ${(k - 2)}^{2}$

choices for (z, t). The forbidden equality z = t occurs when the common color is different from 2, x, and y, giving k − 3 forbidden choices. Therefore, the number of valid choices for (z, t) is

{(k - 2)}^{2} - (k - 3) = k^{2} - 5 k + 7 .

Thus, the contribution of this case is

(k - 2) (k - 3) (k^{2} - 5 k + 7) .

Combining the two cases, we obtain

N_{diff} (k) = 2 {(k - 2)}^{3} + (k - 2) (k - 3) (k^{2} - 5 k + 7) .

Expanding gives

\begin{matrix} N_{diff} (k) & = 2 {(k - 2)}^{3} + (k - 2) (k - 3) (k^{2} - 5 k + 7) \\ = k^{4} - 8 k^{3} + 26 k^{2} - 41 k + 26 . \end{matrix}

Therefore

N_{diff} (k) = ψ (k),

where

ψ (k) = k^{4} - 8 k^{3} + 26 k^{2} - 41 k + 26 .

The enumeration is summarized in Table 1.

Table 1. Case analysis for the local extension count $N_{diff} (k) = ψ (k)$ .

Case	Choices for $(x, y)$	Choices for $(z, t)$	Contribution
$(x, y)$ contains color $2$	$2 (k - 2)$	${(k - 2)}^{2}$	$2 {(k - 2)}^{3}$
$(x, y)$ does not contain color $2$	$(k - 2) (k - 3)$	$k^{2} - 5 k + 7$	$(k - 2) (k - 3) (k^{2} - 5 k + 7)$

As an independent algebraic verification, direct symbolic computation gives

P_{3} (k) = ψ (k) P_{2} (k),

which confirms the consistency of this direct combinatorial derivation.

Derivation of the recurrence and closed-form expression

3.3.

Let $P_{n} (k)$ denote the chromatic polynomial of $G_{n}$ . By Lemma 3.1.1, every proper $k$ -coloring of $G_{n - 1}$ can be extended to the new block in exactly $N_{diff} (k)$ ways. From the direct enumeration in Section 3.2 ,

N_{diff} (k) = ψ (k),

where

ψ (k) = k^{4} - 8 k^{3} + 26 k^{2} - 41 k + 26 .

Hence, for $n \geq 3$

P_{n} (k) = N_{diff} (k) P_{n - 1} (k) = ψ (k) P_{n - 1} (k) .

Since both sides are polynomials in $k$ and the equality holds for all positive integers $k$ , the recurrence holds as a polynomial identity.

Applying this recurrence repeatedly gives

P_{n} (k) = {[ψ (k)]}^{n - 2} P_{2} (k), n \geq 2 .

This proves both the first-order recurrence and the closed-form expression for

F_{n} \times P_{2}

4. Results

4.1 Structural analysis of $F_{n} \times P_{2}$

Theorem 4.1.1:

(Structural Properties of $F_{n} \times P_{2}$ )

For $n \geq 2$ , the graph $F_{n} \times P_{2}$ possesses the following structural properties, which directly affect its chromatic behavior:

1. $| V | = 4 n + 2 .$
2. $| E | = 8 n + 1$ .
3. Degree sequence ${{(2 n + 1)}^{(2)}, 3^{(4 n)}}$ .

Where the notation $d^{(η)}$ denotes $(η)$ vertices of degree $d$ .

Proof:

1. Vertex count: The friendship graph $F_{n}$ has $(2 n + 1)$ vertices. Since $P_{2}$ has two vertices, the Cartesian product has
$| V (F_{n} \times P_{2}) | = 2 (2 n + 1) = 4 n + 2 .$
2. Edge count: The graph $F_{n} \times P_{2},$ consists of two copies of $F_{n}$ , giving
$2 | E (F_{n}) | = 2 (3 n) = 6 n$
internal edges. In addition, there is one vertical edge joining the two copies of each vertex of $F_{n}$ , giving
$| V (F_{n}) | = 2 n + 1$
vertical edges. Therefore,
$| E (F_{n} \times P_{2}) | = 6 n + (2 n + 1) = 8 n + 1 .$
3. Connectivity: Since $F_{n}$ is connected and $P_{2}$ is connected, their Cartesian product $F_{n} \times P_{2}$ is connected. Therefore, $F_{n} \times P_{2}$ is connected for all $n \geq 2$ .

4. Degree sequence: There are two central vertices, one in each layer. Each central vertex is adjacent to $2 n$ peripheral vertices in its own layer and to the corresponding central vertex in the other layer. Hence each central vertex has degree

2 n + 1 .

There are $4 n$ peripheral vertices in total. Each peripheral vertex is adjacent to one central vertex in its layer, one partner vertex in the same triangle, and its corresponding vertex in the other layer. Hence each peripheral vertex has degree $3 .$

Therefore, the degree sequence is

{{(2 n + 1)}^{(2)}, 3^{(4 n)}} . ∎

Remark 4.1.2:

(Edge classification in $F_{n} \times P_{2}$ )

The edge count in Theorem 4.1.1 can be refined by classifying the edges of $F_{n} \times P_{2}$ into three types, as shown in Table 2. There is one central edge joining the two central vertices. The center-peripheral edges consist of the $2 n$ edges incident with the center in each layer, giving $4 n$ edges in total. The remaining peripheral edges consist of the $2 n$ triangle-base edges in the two layers and the $2 n$ vertical edges between corresponding peripheral vertices, giving $4 n$ edges. Hence,

Table 2. Edge classification in $F_{n} \times P_{2}$ .

Edge type	Description	Degree of endpoints	Count
Central edge	Connects the two central vertices $v_{0}^{A}$ and $v_{0}^{B}$	$(2 n + 1, 2 n + 1)$	1
Center-peripheral edges	Connect central vertices to peripheral vertices within each layer	$(2 n + 1, 3)$	$4 n$
Peripheral-peripheral edges	Include triangle-base edges and peripheral vertical edges	$(3, 3)$	$4 n$

1 + 4 n + 4 n = 8 n + 1,

which agrees with the edge count in Theorem 4.1.1.

4.2 Chromatic polynomial analysis

Theorem 4.2.1:

(Recurrence Relation)

For all $n \geq 3$ , the chromatic polynomial of $F_{n} \times P_{2}$ satisfies a first-order linear recurrence

P_{n} (k) = ψ (k) P_{n - 1} (k),

where

ψ (k) = k^{4} - 8 k^{3} + 26 k^{2} - 41 k + 26 .

Proof:

By Lemma 3.1.1, once the colors of the two central vertices are fixed, the number of admissible extensions to the newly attached block $B_{n}$ is independent of the coloring of $G_{n - 1}$ . Hence,

P_{n} (k) = N_{diff} (k) P_{n - 1} (k) .

By the direct combinatorial enumeration in Section 3.2,

N_{diff} (k) = ψ (k) .

Therefore,

P_{n} (k) = ψ (k) P_{n - 1} (k),

as required. ∎

Theorem 4.2.2:

(Closed-Form Expression)

For $n \geq 2$ , the chromatic polynomial of $F_{n} \times P_{2}$ is given by

P_{n} (k) = {[ψ (k)]}^{n - 2} P_{2} (k) .

Proof:

We prove the formula by induction on $n$ . For $n = 2$ , we have

P_{2} (k) = {[ψ (k)]}^{0} P_{2} (k),

so, the formula holds.

Assume that the formula holds for $n - 1$ , where $n \geq 3$ . That is,

P_{n - 1} (k) = {[ψ (k)]}^{n - 3} P_{2} (k) .

Using the recurrence relation from Theorem 4.2.1,

P_{n} (k) = ψ (k) P_{n - 1} (k),

we obtain

P_{n} (k) = ψ (k) {[ψ (k)]}^{n - 3} P_{2} (k) = {[ψ (k)]}^{n - 2} P_{2} (k) .

Therefore, the formula holds for all $n \geq 2$ . ∎

Corollary 4.2.3

(Computational Efficiency for Fixed $k$ )

For a fixed numerical value of $k$ , the closed-form expression

P_{n} (k) = {[ψ (k)]}^{n - 2} P_{2} (k)

allows $P_{n} (k)$ to be evaluated using $O (log n)$ arithmetic multiplications by applying exponentiation by squaring to ${[ψ (k)]}^{n - 2}$ .

This complexity statement concerns numerical evaluation at a fixed value of k. It should not be interpreted as a bound for computing the fully expanded symbolic polynomial $P_{n} (k)$ , since the degree and output size of the polynomial increase with $n$ Indeed,

deg P_{n} (k) = deg P_{2} (k) + (n - 2) deg ψ (k) .

Since $deg P_{2} (k) = 10$ and $deg ψ (k) = 4$ , we obtain

deg P_{n} (k) = 10 + 4 (n - 2) = 4 n + 2 .

Thus, the closed-form expression gives an efficient method for fixed- $k$ numerical evaluation, whereas the fully expanded symbolic polynomial has size growing with $n$ .

Proposition 4.2.4

(Chromatic Number)

For all $n \geq 2$ ,

χ (F_{n} \times P_{2}) = 3

Proof:

Since $F_{n} \times P_{2}$ contains copies of $C_{3}$ , at least three colors are required. Hence,

χ (F_{n} \times P_{2}) \geq 3 .

It remains to show that three colors are sufficient. Define a coloring $c : V (F_{n} \times P_{2}) \to {1, 2, 3}$ as follows:

c (v_{0}^{A}) = 1, c (v_{0}^{B}) = 2 .

For each $i = 1, \dots, n$ , assign

c (u_{i}^{A}) = 2, c (w_{i}^{A}) = 3,

and

c (u_{i}^{B}) = 3, c (w_{i}^{B}) = 1 .

In layer $A$ , each triangle $(v_{0}^{A}, u_{i}^{A}, w_{i}^{A})$ receives the three distinct colors $1, 2, 3$ . In layer $B$ , each triangle $(v_{0}^{B}, u_{i}^{B}, w_{i}^{B})$ receives the three distinct colors $2, 3, 1$ . The central vertical edge satisfies

c (v_{0}^{A}) \neq c (v_{0}^{B}),

and the peripheral vertical edges satisfy

c (u_{i}^{A}) \neq c (u_{i}^{B}), c (w_{i}^{A}) \neq c (w_{i}^{B})

for every

i

. Therefore, all adjacent vertices receive distinct colors, so

c

is a proper 3-coloring. Hence,

χ (F_{n} \times P_{2}) \leq 3 .

Combining the lower and upper bounds gives

χ (F_{n} \times P_{2}) = 3 . ∎

4.3 Algebraic and asymptotic analysis

Theorem 4.3.1

(Real Roots of $ψ (k))$

The chromatic transition polynomial

ψ (k) = k^{4} - 8 k^{3} + 26 k^{2} - 41 k + 26

has exactly two real roots. One root is $k = 2$ , and the other lies in the interval $(2,2.5)$ . In particular, both real roots lie in $[2, 3]$ .

Proof:

Direct evaluation gives

$ψ (2) = 0$ , $ψ (2.5) = 0.0625 > 0$ , $ψ (3) = 2 > 0$ .

The first and second derivatives are

ψ^{'} (k) = 4 k^{3} - 24 k^{2} + 52 k - 41,

and

ψ^{''} (k) = 12 k^{2} - 48 k + 52 .

The discriminant of $ψ^{''} (k)$ is

{(- 48)}^{2} - 4 (12) (52) = - 192 < 0 .

Since the leading coefficient of $ψ^{''} (k)$ is positive, it follows that

ψ^{''} (k) > 0

for all real

k

. Hence

ψ^{'} (k)

is strictly increasing on

ℝ

, and

ψ (k)

is strictly convex.

Moreover,

ψ^{'} (2) = - 1 < 0, ψ^{'} (2.5) = 1.5 > 0 .

Therefore, $ψ^{'} (k)$ has a unique zero $α \in (2,2.5)$ . Thus, $ψ (k)$ is strictly decreasing on $(- \infty, α)$ and strictly increasing on $(α, \infty)$ .

Since $ψ (2) = 0$ and $ψ^{'} (2) < 0$ , the polynomial becomes negative immediately to the right of $2$ . On the other hand, $ψ (2.5) > 0$ . Hence, by the Intermediate Value Theorem, $ψ (k)$ has a root $ρ \in (2,2.5)$ .

Finally, because $ψ (k)$ is decreasing before $α$ and increasing after $α$ , it can have at most one real root on each side of $α$ . Since $k = 2$ is one root and $ρ \in (2,2.5)$ is another, these are the only real roots of $ψ (k)$ . The remaining two roots are non-real complex conjugates. ∎

Theorem 4.3.2

(Exponential Growth Rate)

Let $k \in ℂ$ be fixed and suppose that $P_{2} (k) \neq 0$ . Then

lim_{n \to \infty} | P_{n} (k) |^{\frac{1}{n}} = | ψ (k) | .

In particular, for every integer $k \geq 3$ ,

lim_{n \to \infty} P_{n} {(k)}^{\frac{1}{n}} = ψ (k) .

Proof:

From the closed-form expression in Theorem 4.2.2, we have

P_{n} (k) = {[ψ (k)]}^{n - 2} P_{2} (k) (n \geq 2) .

Taking absolute values gives

| P_{n} (k) | = | ψ (k) |^{n - 2} | P_{2} (k) | .

Hence,

| P_{n} (k) |^{\frac{1}{n}} = | ψ (k) |^{\frac{(n - 2)}{n}} | P_{2} (k) |^{\frac{1}{n}} .

Since $\frac{n - 2}{n} \to 1$ and, because $P_{2} (k) \neq 0$ ,

| P_{2} (k) |^{\frac{1}{n}} \to 1,

we obtain

lim_{n \to \infty} | P_{n} (k) |^{\frac{1}{n}} = | ψ (k) | .

For integer $k \geq 3,$ $P_{n} (k)$ counts proper $k$ -colorings, so $P_{n} (k) > 0$ . Also, $ψ (k) = N_{diff} (k) > 0$ , since it counts the number of admissible extensions to one additional block. Therefore

| P_{n} (k) | = P_{n} (k), | ψ (k) | = ψ (k),

and consequently

lim_{n \to \infty} P_{n} {(k)}^{\frac{1}{n}} = ψ (k) .

This proves the result. ∎

Remark 4.3.3

(Absolute Values and the Combinatorial Range)

The absolute values are necessary when $k$ is treated as a real or complex parameter, since $P_{n} (k)$ may be negative or complex-valued. The simpler expression

lim_{n \to \infty} P_{n} {(k)}^{\frac{1}{n}} = ψ (k) .

is valid in the combinatorial range

k \in ℤ

k \geq 3

, where both

P_{n} (k)

and

ψ (k)

are positive.

Corollary 4.3.4

(Ratio Convergence of Chromatic Polynomials)

For every fixed $k$ such that $ψ (k) \neq 0$ and $P_{2} (k) \neq 0$ ,

lim_{n \to \infty} \frac{P_{n} (k)}{P_{n - 1} (k)} = ψ (k) .

Proof:

By Theorem 4.2.1,

P_{n} (k) = ψ (k) P_{n - 1} (k), n \geq 3 .

Since

ψ (k) \neq 0

and

P_{2} (k) \neq 0

, the closed-form expression implies

P_{n - 1} (k) \neq 0

for all

n \geq 3

Hence

\frac{P_{n} (k)}{P_{n - 1} (k)} = ψ (k),

and the stated limit follows immediately. ∎

Interpretation

The value $| ψ (k) |$ is the exponential growth constant of the sequence ${P_{n} (k)}$ for fixed $k$ with $P_{2} (k) \neq 0$ . In the combinatorial range $k \in ℤ$ , $k \geq 3$ , the recurrence gives the exact relation

P_{n} (k) = ψ (k) P_{n - 1} (k),

so, each additional block multiplies the number of proper

k

-colorings by

ψ (k)

Remark 4.3.5

(Root distribution)

From the closed-form expression in Theorem 4.2.2, we have

P_{n} (k) = {[ψ (k)]}^{n - 2} P_{2} (k) .

Therefore, the zero set of

P_{n} (k)

is contained in the union of the zero set of

P_{2} (k)

and the zero set of

ψ (k)

. More precisely, if

r

is a root of

ψ (k)

of multiplicity

m_{ψ} (r)

and a root of

P_{2} (k)

of multiplicity

m_{2} (r)

, then its multiplicity as a root of

P_{n} (k)

m_{2} (r) + (n - 2) m_{ψ} (r) .

If $r$ is a root of $ψ (k)$ but not a root of $P_{2} (k)$ , then its multiplicity in $P_{n} (k)$ is $(n - 2) m_{ψ} (r)$ . Hence, as $n$ increases, the locations of the roots remain in a fixed finite set, while the multiplicities of the roots contributed by $ψ (k)$ grow linearly with $n$ .

4.4 Numerical validation

To verify the recurrence relation $P_{n} (k) = ψ (k) P_{n - 1} (k),$ from Theorem 4.2.1 and the closed-form expression $P_{n} (k) = {[ψ (k)]}^{n - 2} P_{2} (k)$ from Theorem 4.2.2, we computed the chromatic polynomials $P_{n} (k)$ for $n = 2, 3, 4, 5$ using Wolfram Mathematica.

For the tested integer values $k = 3, 4, 5, 6$ , the computations showed agreement among the following three approaches:

• direct computation of $P_{n} (k)$ ;
• evaluation using the recurrence relation;
• evaluation using the closed-form expression.

The numerical values of the transition polynomial $ψ (k)$ and the chromatic polynomials $P_{2} (k)$ , $P_{3} (k)$ , $P_{4} (k)$ , and $P_{5} (k)$ are listed in Table 3.

Table 3. Numerical values of $P_{n} (k)$ for $n = 2, 3, 4, and 5$ , along with $ψ (k)$ at integer values $k \geq 3$ .

$k$	$ψ (k)$	$P_{2} (k)$	$P_{3} (k)$	$P_{4} (k)$	$P_{5} (k)$
3	2	24	48	96	192
4	22	5,808	127,776	2,811,072	61,843,584
5	96	184,320	17,694,720	1,698,693,120	163,074,539,520
6	284	2,419,680	687,189,120	195,161,710,080	55,425,925,662,720

The following sample calculations illustrate the agreement:

• Theorem 4.2.1: $P_{3} (3) = ψ (3) P_{2} (3) = 2 \cdot 24 = 48 .$
• Theorem 4.2.2: $P_{5} (6) = {[ψ (6)]}^{3} P_{2} (6) = 284^{3} \cdot 2,419,680 = 55,425,925,662,720 .$

Moreover, the recurrence gives the exact growth ratio for fixed $k$ :

\frac{P_{5} (4)}{P_{4} (4)} = \frac{61,843,584}{2,811,072} = 22 = ψ (4) .

This exact ratio should be distinguished from the asymptotic $n$ -th root behavior established in Theorem 4.3.2. In the positive integer coloring range $k \geq 3$ , that theorem gives

lim_{n \to \infty} P_{n} {(k)}^{\frac{1}{n}} = ψ (k) .

For finite $n$ , however, the quantity $P_{n} {(k)}^{\frac{1}{n}}$ need not be close to $ψ (k)$ . For example,

P_{5} {(4)}^{\frac{1}{5}} = {(61,843,584)}^{\frac{1}{5}} \approx 36.16,

whereas

ψ (4) = 22 .

The explicit polynomial expressions used as the basis for these computations are:

P_{2} (k) = k^{10} - 17 k^{9} + 132 k^{8} - 614 k^{7} + 1882 k^{6} - 3932 k^{5} + 5581 k^{4} - 5165 k^{3} + 2808 k^{2} - 676 k .

P_{3} (k) = k^{14} - 25 k^{13} + 294 k^{12} - 2153 k^{11} + 10949 k^{10} - 40806 k^{9} + 114575 k^{8} - 245171 k^{7} + 399378 k^{6} - 488483 k^{5} + 435287 k^{4} - 266994 k^{3} + 100724 k^{2} - 17576 k .

These computations provide computational confirmation of Theorem 4.2.1 and Theorem Theorem 4.2.2 for the tested values. They also illustrate the ratio identity $\frac{P_{n} (k)}{P_{n - 1} (k)} = ψ (k)$ , which is an exact consequence of the recurrence for $n \geq 3$ . The asymptotic growth statement remains the limit result established in Theorem 4.3.2.

4.5 Illustrative application: a two-period scheduling model

This section presents a two-period scheduling interpretation in which the derived chromatic polynomial counts feasible room assignments under explicitly stated conflict constraints. The model is formulated under homogeneous-room and pairwise-conflict assumptions, providing a controlled resource-allocation setting for interpreting the recurrence relation and the closed-form expression.

4.5.1 Model specification

Consider a conference with one coordinator and $n$ independent teams. The schedule is divided into two consecutive periods, denoted by $A$ and $B$ . The available resources are $k$ identical meeting rooms.

The graph $G_{n} = F_{n} \times P_{2}$ is used as a conflict graph. Each vertex represents one session requiring a room, and each edge represents a pair of sessions that cannot be assigned to the same room. Thus, a proper $k$ -coloring of $G_{n}$ corresponds to a valid assignment of $k$ rooms to all sessions.

4.5.2 Precise vertex-to-session mapping

The vertex set of $G_{n}$ is

V (G_{n}) = {v_{0}^{A}, v_{0}^{B}} \cup {u_{i}^{A}, w_{i}^{A}, u_{i}^{B}, w_{i}^{B} : 1 \leq i \leq n} .

The two central vertices

v_{0}^{A}, v_{0}^{B}

represent the coordinator sessions in periods $A$ and $B$ , respectively.

For each team $i$ , the vertices

u_{i}^{A}, w_{i}^{A}

represent two team-related sessions in period $A$ , while

u_{i}^{B}, w_{i}^{B}

represent the corresponding two team-related sessions in period

B

. Therefore, the total number of sessions is

2 + 4 n = 4 n + 2,

which agrees exactly with the vertex count of

F_{n} \times P_{2}

4.5.3 Edge-to-conflict mapping

The edge set of $G_{n}$ encodes the scheduling constraints as follows.

For each $i = 1, \dots, n$ and each period $X \in {A, B}$ , the edges

v_{0}^{X} u_{i}^{X}, v_{0}^{X} w_{i}^{X}

represent coordinator-team conflicts within the same period. These sessions cannot be assigned to the same room.

The edge

u_{i}^{X} w_{i}^{X}

represents a conflict between the two team-related sessions of team

i

in the same period.

The vertical edges

v_{0}^{A} v_{0}^{B}, u_{i}^{A} u_{i}^{B}, w_{i}^{A} w_{i}^{B}

encode cross-period constraints between corresponding sessions. Under this modeling assumption, such paired sessions must receive distinct room assignments.

Thus, feasible schedules are exactly the proper $k$ -colorings of $G_{n}$ . Consequently, $P_{n} (k) = P (G_{n}, k)$ counts the number of feasible room assignments using $k$ rooms.

4.5.4 Interpretation of scheduling parameters

Within the proposed two-period scheduling model, the parameter $k$ denotes the number of available rooms and is therefore restricted to positive integer values. The feasibility of the scheduling problem is governed by the chromatic number of the conflict graph. Since

χ (F_{n} \times P_{2}) = 3,

the minimum feasible number of rooms in this model is

k_{\min} = 3 .

Thus, three rooms are necessary and sufficient to obtain a valid room assignment for every $n \geq 2$ .

2. The transition polynomial

ψ (k) = k^{4} - 8 k^{3} + 26 k^{2} - 41 k + 26

plays a different role. By the recurrence relation

P_{n} (k) = ψ (k) P_{n - 1} (k)

, we have, for fixed

k

\frac{P_{n} (k)}{P_{n - 1} (k)} = ψ (k) .

Hence, $ψ (k)$ measures the multiplicative factor in the number of feasible schedules when one additional team is added, while the number of available rooms is kept fixed. For example,

ψ (3) = 2, ψ (4) = 22, ψ (5) = 96 .

Accordingly, with $k = 4$ rooms fixed, increasing the model from $n - 1$ to $n$ teams multiplies the number of feasible schedules by 22. With $k = 5$ rooms fixed, the corresponding multiplicative factor is 96.

For a fixed conference size $n$ , the effect of changing the number of rooms is quantified by comparing the chromatic polynomial at different room numbers:

R_{n} (k_{2}, k_{1}) = \frac{P_{n} (k_{2})}{P_{n} (k_{1})}, k_{2} > k_{1} .

Thus, the recurrence factor $ψ (k)$ describes scalability with respect to the number of teams for fixed $k$ , whereas the ratio $R_{n} (k_{2}, k_{1})$ describes the change in scheduling flexibility obtained by increasing the number of available rooms for fixed $n$ .

4.5.5 Quantitative illustration

We now illustrate the two distinct quantitative effects described above. The first concerns the growth in the number of feasible schedules when the number of teams increases while the number of rooms is fixed. The second concerns the change in scheduling flexibility when the number of rooms increases while the number of teams is fixed.

Scheduling with the minimum number of rooms ( $k = 3$ ).

At the minimum feasible room count, we have $ψ (3) = 2$ . By the recurrence relation,

P_{n} (3) = 2 P_{n - 1} (3) .

Thus, with three rooms, each additional team doubles the number of feasible schedules. Table 4 gives the corresponding values.

Table 4. Scheduling with the minimum number of rooms $(k = 3)$ .

Conference scale	Teams $n$	Valid schedules $P_{n} (3)$	Growth from previous $n$
Small conference	3	48	—
Medium conference	4	96	$\times 2$
Medium conference	5	192	$\times 2$
Large conference	6	384	$\times 2$

Scalability with respect to the number of teams

To illustrate the effect of adding one team while keeping the room count fixed, we compare $P_{5} (k)$ and $P_{4} (k)$ . The ratios are shown in Table 5, and they coincide exactly with $ψ (k)$ .

Table 5. Scalability for fixed room availability.

Number of rooms $k$	$P_{4} (k)$	$P_{5} (k)$	$\frac{P_{5} (k)}{P_{4} (k)}$	Interpretation
3	96	192	$ψ (3) = 2$	Adding one team doubles the schedules
4	2,811,072	61,843,584	$ψ (4) = 22$	Adding one team multiplies schedules by $22$
5	1,698,693,120	163,074,539,520	$ψ (5) = 96$	Adding one team multiplies schedules by $96$

The ratios in Table 5 compare $P_{5} (k)$ with $P_{4} (k)$ for the same value of $k$ ; they therefore describe the effect of adding one team under fixed room availability. By contrast, the effect of increasing the number of rooms is measured by comparing $P_{n} (k)$ at different values of $k$ for a fixed value of $n$ .

Effect of increasing the number of rooms

For a fixed conference size $n$ , the effect of increasing the number of available rooms is measured by comparing $P_{n} (k)$ at different values of $k$ . Table 6 illustrates this effect for $n = 5$ , using $k = 3$ as the baseline.

Table 6. Effect of increasing the number of rooms for fixed $n = 5$ .

Number of rooms $k$	$P_{5} (k)$	Flexibility ratio relative to $k = 3$
3	192	$1$
4	61,843,584	$322,102$
5	163,074,539,520	$849,346,560$

For instance,

\frac{P_{5} (4)}{P_{5} (3)} = \frac{61,843,584}{192} = 322,102 .

Thus, for n = 5, increasing the number of rooms from $3$ to $4$ gives a flexibility ratio of $322,102$ Similarly,

\frac{P_{5} (5)}{P_{5} (3)} = \frac{163,074,539,520}{192} = 849,346,560 .

In comparison, the value $ψ (4) = 22$ is the fixed-room growth factor from $P_{4} (4)$ to $P_{5} (4)$ , namely

\frac{P_{5} (4)}{P_{4} (4)} = 22 .

4.5.6 Summary of the Graph-Scheduling Correspondence

The correspondence between the graph model and the scheduling interpretation is summarized in Table 7.

Table 7. From Graph Elements to Scheduling Interpretation.

Graph-theoretic element	Scheduling interpretation	Count	Meaning
$G_{n} = F_{n} \times P_{2}$	Two-period simplified scheduling model	—	Conflict graph for room assignment
$v_{0}^{A}, v_{0}^{B}$	Coordinator sessions in periods $A$ and $B$	$2$	Central scheduling sessions
$u_{i}^{A}, w_{i}^{A}$	Two team-related sessions of team $i$ in period $A$	$2 n$	Team sessions in period $A$
$u_{i}^{B}, w_{i}^{B}$	Two team-related sessions of team $i$ in period $B$	$2 n$	Team sessions in period $B$
All vertices	Sessions requiring rooms	$4 n + 2$	Total number of scheduled sessions
$v_{0}^{X} u_{i}^{X}, v_{0}^{X} w_{i}^{X}$	Coordinator-team conflicts in period $X$	$4 n$	These sessions cannot share a room
$u_{i}^{X} w_{i}^{X}$	Within-team conflict in period $X$	$2 n$	The two team sessions require different rooms
$u_{i}^{A} u_{i}^{B}, w_{i}^{A} w_{i}^{B}$	Cross-period team constraints	$2 n$	Corresponding sessions cannot reuse the same room
$v_{0}^{A} v_{0}^{B}$	Coordinator cross-period constraint	$1$	Coordinator sessions cannot reuse the same room
All edges	Room-conflict constraints	$8 n + 1$	Total number of conflict constraints
Proper $k$ -coloring	Valid room assignment using $k$ rooms	$P_{n} (k)$	Counts all feasible schedules
$χ (G_{n}) = 3$	Minimum feasible number of rooms	$3$	Practical feasibility threshold
$ψ (k)$	Fixed- $k$ team-growth factor	—	Growth factor when one team is added

4.5.7 Model Limitations and Assumptions

This scheduling interpretation is formulated under a controlled set of assumptions that make the correspondence between graph colorings and feasible room assignments explicit. It assumes that:

1. all teams have the same conflict structure;
2. all rooms are homogeneous and interchangeable;
3. the only constraints are those encoded by the edges of $F_{n} \times P_{2}$ ;
4. cross-period constraints require corresponding sessions to use different rooms;
5. no room capacities, time durations, preferences, or probabilistic constraints are included.

These assumptions delimit the scope of the model while preserving its role as a quantitative interpretation of the chromatic polynomial in a two-period resource-allocation setting.

5. Discussion

Theoretical contribution

5.1.

This work provides an explicit algebraic characterization of the chromatic polynomial of $F_{n} \times P_{2}$ , a Cartesian product graph family with a layered triangular structure. The main result is the first-order recurrence $P_{n} (k) = ψ (k) P_{n - 1} (k), n \geq 3,$ together with the closed-form expression $P_{n} (k) = {[ψ (k)]}^{n - 2} P_{2} (k), n \geq 2,$ where $ψ (k)$ is the degree-four transition polynomial derived in Section 3.2.

The key structural reason for this formula is the conditional independence of the recursively attached block $B_{n}$ , as formalized in Lemma 3.1.1. Once the colors of the two central vertices are fixed and distinct, the four new peripheral vertices form a local extension problem whose number of admissible colorings depends only on $k$ . Thus, the global chromatic polynomial is controlled by repeated copies of the same local transition count.

This result complements existing work on graph products and coloring⁶^,⁷ and on related structured families.⁸^–¹² Compared with earlier Cartesian-product families such as $P_{2} \times P_{n}$ , $C_{3} \times P_{n}$ , $C_{3} □ P_{n}$ , and $P_{3} □ P_{n}$ , the present family has a different recursive mechanism: the parameter $n$ increases the number of friendship triangles sharing a common center rather than extending a path direction. This shared-center structure reduces the chromatic recurrence to a single degree-four transition polynomial, reflecting the four new peripheral vertices added at each recursive step.

Structural, computational, and scheduling implications

5.2

The structural results show that

| V | = 4 n + 2, | E | = 8 n + 1, χ (F_{n} \times P_{2}) = 3 .

Hence, the graph grows linearly in size while its chromatic number remains constant. This separates chromatic feasibility from enumerative growth: three colors are sufficient for all $n \geq 2$ , but the number of proper colorings grows according to the transition polynomial $ψ (k)$ .

The closed-form expression also has a direct computational consequence. For fixed numerical $k$ , the value $P_{n} (k)$ can be evaluated efficiently from the closed form using exponentiation by squaring in $O (log n)$ arithmetic multiplications. This applies to fixed- $k$ numerical evaluation, not to full symbolic expansion, since

deg P_{n} (k) = 4 n + 2 .

Thus, the formula is both algebraically explicit and computationally useful for evaluating large members of the family.

The two-period scheduling interpretation gives a quantitative meaning to the same polynomial. In that model, $P_{n} (k)$ counts feasible room assignments under the stated conflict constraints. The chromatic number gives the minimum feasible number of rooms, $k_{\min} = 3,$ while

\frac{P_{n} (k)}{P_{n - 1} (k)} = ψ (k)

measures the growth in the number of feasible schedules when one additional team is added, and the number of rooms is kept fixed. The effect of increasing the number of rooms for fixed

n

is instead measured by ratios of the form

\frac{P_{n} (k_{2})}{P_{n} (k_{1})}, k_{2} > k_{1} .

This distinction gives the scheduling model a precise combinatorial interpretation while remaining within the stated scope of the model.

Limitations and future directions

5.3

The present work focuses on $F_{n} \times P_{2}$ . A natural extension is $F_{n} \times P_{m}$ for $m > 2$ , which would correspond to a multi-period layered model and may require multiple boundary-coloring states rather than a single transition polynomial.

Another direction is to incorporate heterogeneous resources using list-coloring or weighted-coloring models. Such extensions could represent room capacities, team-specific requirements, or restricted availability of resources.

Finally, since the roots of $P_{n} (k)$ are contained in the union of the zero sets of $P_{2} (k)$ and $ψ (k)$ , as described in Remark 4.3.5, this family may provide a tractable setting for studying chromatic-root multiplicities and their relation to partition-function phenomena such as the Potts model.¹⁶

Ethical considerations

This study does not involve human participants, animal subjects, or sensitive data. Therefore, no ethical approval was required.

Use of AI-assisted technology

During manuscript revision, AI-assisted tools, including ChatGPT and DeepSeek, were used as supplementary aids for language editing, formatting support, and algebraic-verification checks. The authors critically reviewed all outputs and take full responsibility for the accuracy, originality, and integrity of the final manuscript.

Data availability

This is a theoretical study in algebraic graph theory. All results, including the recurrence relation, the closed-form expression for the chromatic polynomial, and all numerical values, are derived analytically and presented within the article. No external datasets were generated or analyzed. All findings are fully reproducible using the formulas and methods provided in Sections 3 and 4.

Reporting guidelines

This is a theoretical mathematical study and does not involve clinical trials, animal experiments, observational studies, or qualitative research. Therefore, no specific reporting guidelines (e.g., CONSORT, ARRIVE, STROBE, COREQ) are applicable.

Acknowledgements

The authors thank Tikrit University for providing academic support and resources.

References

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Author details Author details

¹ Mathematics, Tikrit University, Tikrit, Saladin Governorate, 34001, Iraq

Sarah M. Talab
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Resources, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

Nabeel E. Arif
Roles: Supervision

Competing interests

No competing interests were disclosed.

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The author(s) declared that no grants were involved in supporting this work.

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Reviewer Report 11 May 2026

Siti Amiroch, Universitas Islam Darul ‘ulum, Lamongan, Indonesia

Approved with Reservations

https://doi.org/10.5256/f1000research.195018.r474977

Reviewer Report

The manuscript studies the chromatic polynomial of the graph family FnxP2

, where Fn

denotes the friendship graph and P2

is the path on two vertices. The topic is relevant to algebraic and combinatorial graph theory, and the ... Continue reading

Reviewer Report

The manuscript studies the chromatic polynomial of the graph family FnxP2

, where Fn

denotes the friendship graph and P2

is the path on two vertices. The topic is relevant to algebraic and combinatorial graph theory, and the recursive structure considered in the paper is potentially useful. The manuscript is generally organised in a logical sequence and the numerical examples help illustrate the proposed recurrence relation.
General Evaluation
The main contribution of the paper is the derivation of a recurrence relation and a closed-form expression for the chromatic polynomial of FnxP2

, together with structural observations, chromatic number analysis, asymptotic interpretation, and a scheduling application. The recursive approach is promising. However, several substantive issues require correction before the manuscript can be considered suitable for publication in a high-quality Scopus-indexed journal.
Strengths
1. The paper addresses a specialised and relevant family of Cartesian product graphs.
2. The recursive decomposition is a promising approach for obtaining a compact chromatic polynomial formula.
3. The chromatic number result is useful and consistent with the triangular structure of the graph.
4. The numerical examples support the recurrence formula for the tested cases.
5. The attempt to connect the algebraic results with scheduling applications gives the paper potential applied relevance.

Main Comments and Recommendations
1. Section 3.2 — Derivation of the transition polynomial
The derivation of the transition polynomial ψ(k) requires substantial clarification. The coefficients in
ψk=k4-8k3+26k2-41k+26

are stated without a complete enumeration of the corresponding inclusion–exclusion terms. Since this polynomial is the central component of the recurrence and closed-form formula, the manuscript should provide either a full counting table, a structured case analysis, or an appendix containing the complete derivation.
2. Section 4.1 — Claim on non-planarity
The assertion that FnxP2

is non-planar for n ≥ 3 is not sufficiently justified. The manuscript states that the graph contains a K3,3

minor, but no explicit construction of such a minor is provided. This is a substantial mathematical claim and should either be proved rigorously by specifying the required branch sets or revised if the claim does not hold.
3. Section 4.2 — Computational complexity
The statement that the closed-form expression reduces the computation of Pn(k)

to O(log n) is too broad. This claim is valid only for numerical evaluation at a fixed value of k using exponentiation by squaring. It does not apply to symbolic expansion of the polynomial, since the degree and output size increase with n. The statement should be qualified accordingly.
4. Section 4.3 — Asymptotic growth
The asymptotic statement involving Pn(k)1n

should be formulated more carefully. For general real or complex values of k, the mathematically safer formulation should involve Pn(k)1n

. The current expression is valid only under additional restrictions, for example for integer k ≥ 3. These restrictions should be made explicit.
5. Section 4.5 — Scheduling interpretation
The scheduling interpretation contains two important issues. First, the non-integer real root of ψ(k) should not be interpreted as a theoretical minimum number of rooms. In a scheduling problem, the number of rooms is discrete, and feasibility is determined by the chromatic number χ = 3. Secondly, ψ(4) = 22 is interpreted as the gain obtained by increasing the number of rooms from 3 to 4, whereas ψ(k) represents the growth factor when the number of teams increases while k is fixed. These interpretations should be corrected to avoid overstating the practical implications of the model.
6. Section 4.5 — Mapping between the graph and the scheduling model
The correspondence between FnxP2

and the proposed conference scheduling model is not yet sufficiently precise. The graph has 4n + 2 vertices, but the description involving one coordinator, n teams, two periods, and two tasks per team does not clearly explain how these scheduling entities produce exactly 4n + 2 vertices. A clearer vertex-to-session and edge-to-conflict mapping is required for the application to be convincing.
Final Recommendation
The manuscript has a promising mathematical core, particularly in its recursive treatment of FnxP2

. However, the derivation of the transition polynomial, the structural claim on non-planarity, the asymptotic formulation, and the scheduling interpretation require substantial revision. In its present form, the manuscript is not yet suitable for publication in the journal. A major revision is recommended.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests: No competing interests were disclosed.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

CITE

Report a concern

Author Response 12 Jun 2026

Sarah M. Talab, Mathematics, Tikrit University, Tikrit, 34001, Iraq

12 Jun 2026

Author Response

We sincerely thank the reviewer for the careful evaluation and constructive comments. We appreciate the reviewer’s recognition of the relevance of the topic, the recursive approach, the chromatic number result, ... Continue reading We sincerely thank the reviewer for the careful evaluation and constructive comments. We appreciate the reviewer’s recognition of the relevance of the topic, the recursive approach, the chromatic number result, the numerical verification, and the potential scheduling interpretation. We have revised the manuscript substantially to address all points raised.

Comment 1. Derivation of the transition polynomial.
We agree that the derivation of ψ(k) required more detail. In the revised manuscript, Section 3.2 now includes a structured combinatorial case analysis for the local extension count Ndiff(k). After fixing the colors of the two central vertices, we count the admissible colorings of the four new peripheral vertices and obtain
Ndiff(k)=2(k−2)^3+(k−2)(k−3)(k^2−5k+7)=ψ(k).This gives
ψ(k)=k^4−8k^3+26k^2−41k+26.
A summary table of the enumeration has also been added.

Comment 2. Non-planarity claim.
We agree that the previous non-planarity statement required a rigorous proof. To avoid unsupported claims outside the main focus of the paper, we removed this assertion from the revised manuscript. The structural analysis now focuses on the order, size, connectivity, and degree sequence, which are directly used in the chromatic analysis.

Comment 3. Computational complexity.
We revised the complexity statement. The manuscript now states explicitly that the O(log⁡ n) complexity applies only to fixed-k numerical evaluation using exponentiation by squaring. It does not apply to full symbolic expansion, since
deg⁡Pn(k)=4n+2.

Comment 4. Asymptotic growth.
We reformulated the asymptotic result more carefully. For fixed k with P2(k)≠0, the revised theorem states
limn→∞∣Pn(k)∣^1/n=∣ψ(k)∣
For integer k≥3, where Pn(k) and ψ(k) are positive, this reduces to
lim⁡n→∞Pn(k)^1/n=ψ(k)..

Comment 5. Scheduling interpretation.
We revised the scheduling section to correct the interpretation. The non-integer real root of ψ(k) is no longer interpreted as a minimum number of rooms. The minimum feasible number of rooms is now stated to be determined by the chromatic number:
kmin⁡=3.
We also clarified that ψ(k) measures the growth factor when one additional team is added while the number of rooms is fixed. The effect of increasing the number of rooms is now measured separately by ratios of the form
Pn(k2)/Pn(k1),k2>k1.

Comment 6. Mapping between the graph and the scheduling model.
We expanded the scheduling model by adding a precise vertex-to-session and edge-to-conflict mapping. The revised version explains that the two central vertices represent coordinator sessions in the two periods, while the 4n peripheral vertices represent two team-related sessions for each of the n teams in each of the two periods. Hence the model has exactly
2+4n=4n+2
sessions, matching the vertex count of Fn×P2. The edge constraints are also described explicitly.

We thank the reviewer again for the detailed comments. We believe that the revised manuscript is now more rigorous, clearer, and more precise in its mathematical derivation and scheduling interpretation.
We sincerely thank the reviewer for the careful evaluation and constructive comments. We appreciate the reviewer’s recognition of the relevance of the topic, the recursive approach, the chromatic number result, the numerical verification, and the potential scheduling interpretation. We have revised the manuscript substantially to address all points raised.

Comment 1. Derivation of the transition polynomial.
We agree that the derivation of ψ(k) required more detail. In the revised manuscript, Section 3.2 now includes a structured combinatorial case analysis for the local extension count Ndiff(k). After fixing the colors of the two central vertices, we count the admissible colorings of the four new peripheral vertices and obtain
Ndiff(k)=2(k−2)^3+(k−2)(k−3)(k^2−5k+7)=ψ(k).This gives
ψ(k)=k^4−8k^3+26k^2−41k+26.
A summary table of the enumeration has also been added.

Comment 2. Non-planarity claim.
We agree that the previous non-planarity statement required a rigorous proof. To avoid unsupported claims outside the main focus of the paper, we removed this assertion from the revised manuscript. The structural analysis now focuses on the order, size, connectivity, and degree sequence, which are directly used in the chromatic analysis.

Comment 3. Computational complexity.
We revised the complexity statement. The manuscript now states explicitly that the O(log⁡ n) complexity applies only to fixed-k numerical evaluation using exponentiation by squaring. It does not apply to full symbolic expansion, since
deg⁡Pn(k)=4n+2.

Comment 4. Asymptotic growth.
We reformulated the asymptotic result more carefully. For fixed k with P2(k)≠0, the revised theorem states
limn→∞∣Pn(k)∣^1/n=∣ψ(k)∣
For integer k≥3, where Pn(k) and ψ(k) are positive, this reduces to
lim⁡n→∞Pn(k)^1/n=ψ(k)..

Comment 5. Scheduling interpretation.
We revised the scheduling section to correct the interpretation. The non-integer real root of ψ(k) is no longer interpreted as a minimum number of rooms. The minimum feasible number of rooms is now stated to be determined by the chromatic number:
kmin⁡=3.
We also clarified that ψ(k) measures the growth factor when one additional team is added while the number of rooms is fixed. The effect of increasing the number of rooms is now measured separately by ratios of the form
Pn(k2)/Pn(k1),k2>k1.

Comment 6. Mapping between the graph and the scheduling model.
We expanded the scheduling model by adding a precise vertex-to-session and edge-to-conflict mapping. The revised version explains that the two central vertices represent coordinator sessions in the two periods, while the 4n peripheral vertices represent two team-related sessions for each of the n teams in each of the two periods. Hence the model has exactly
2+4n=4n+2
sessions, matching the vertex count of Fn×P2. The edge constraints are also described explicitly.

We thank the reviewer again for the detailed comments. We believe that the revised manuscript is now more rigorous, clearer, and more precise in its mathematical derivation and scheduling interpretation.
Competing Interests: No competing interests. Close
Report a concern
Respond or Comment

COMMENTS ON THIS REPORT

Author Response 12 Jun 2026

Sarah M. Talab, Mathematics, Tikrit University, Tikrit, 34001, Iraq

12 Jun 2026

Author Response

We sincerely thank the reviewer for the careful evaluation and constructive comments. We appreciate the reviewer’s recognition of the relevance of the topic, the recursive approach, the chromatic number result, ... Continue reading We sincerely thank the reviewer for the careful evaluation and constructive comments. We appreciate the reviewer’s recognition of the relevance of the topic, the recursive approach, the chromatic number result, the numerical verification, and the potential scheduling interpretation. We have revised the manuscript substantially to address all points raised.

Comment 1. Derivation of the transition polynomial.
We agree that the derivation of ψ(k) required more detail. In the revised manuscript, Section 3.2 now includes a structured combinatorial case analysis for the local extension count Ndiff(k). After fixing the colors of the two central vertices, we count the admissible colorings of the four new peripheral vertices and obtain
Ndiff(k)=2(k−2)^3+(k−2)(k−3)(k^2−5k+7)=ψ(k).This gives
ψ(k)=k^4−8k^3+26k^2−41k+26.
A summary table of the enumeration has also been added.

Comment 2. Non-planarity claim.
We agree that the previous non-planarity statement required a rigorous proof. To avoid unsupported claims outside the main focus of the paper, we removed this assertion from the revised manuscript. The structural analysis now focuses on the order, size, connectivity, and degree sequence, which are directly used in the chromatic analysis.

Comment 3. Computational complexity.
We revised the complexity statement. The manuscript now states explicitly that the O(log⁡ n) complexity applies only to fixed-k numerical evaluation using exponentiation by squaring. It does not apply to full symbolic expansion, since
deg⁡Pn(k)=4n+2.

Comment 4. Asymptotic growth.
We reformulated the asymptotic result more carefully. For fixed k with P2(k)≠0, the revised theorem states
limn→∞∣Pn(k)∣^1/n=∣ψ(k)∣
For integer k≥3, where Pn(k) and ψ(k) are positive, this reduces to
lim⁡n→∞Pn(k)^1/n=ψ(k)..

Comment 5. Scheduling interpretation.
We revised the scheduling section to correct the interpretation. The non-integer real root of ψ(k) is no longer interpreted as a minimum number of rooms. The minimum feasible number of rooms is now stated to be determined by the chromatic number:
kmin⁡=3.
We also clarified that ψ(k) measures the growth factor when one additional team is added while the number of rooms is fixed. The effect of increasing the number of rooms is now measured separately by ratios of the form
Pn(k2)/Pn(k1),k2>k1.

Comment 6. Mapping between the graph and the scheduling model.
We expanded the scheduling model by adding a precise vertex-to-session and edge-to-conflict mapping. The revised version explains that the two central vertices represent coordinator sessions in the two periods, while the 4n peripheral vertices represent two team-related sessions for each of the n teams in each of the two periods. Hence the model has exactly
2+4n=4n+2
sessions, matching the vertex count of Fn×P2. The edge constraints are also described explicitly.

We thank the reviewer again for the detailed comments. We believe that the revised manuscript is now more rigorous, clearer, and more precise in its mathematical derivation and scheduling interpretation.
We sincerely thank the reviewer for the careful evaluation and constructive comments. We appreciate the reviewer’s recognition of the relevance of the topic, the recursive approach, the chromatic number result, the numerical verification, and the potential scheduling interpretation. We have revised the manuscript substantially to address all points raised.

Comment 1. Derivation of the transition polynomial.
We agree that the derivation of ψ(k) required more detail. In the revised manuscript, Section 3.2 now includes a structured combinatorial case analysis for the local extension count Ndiff(k). After fixing the colors of the two central vertices, we count the admissible colorings of the four new peripheral vertices and obtain
Ndiff(k)=2(k−2)^3+(k−2)(k−3)(k^2−5k+7)=ψ(k).This gives
ψ(k)=k^4−8k^3+26k^2−41k+26.
A summary table of the enumeration has also been added.

Comment 2. Non-planarity claim.
We agree that the previous non-planarity statement required a rigorous proof. To avoid unsupported claims outside the main focus of the paper, we removed this assertion from the revised manuscript. The structural analysis now focuses on the order, size, connectivity, and degree sequence, which are directly used in the chromatic analysis.

Comment 3. Computational complexity.
We revised the complexity statement. The manuscript now states explicitly that the O(log⁡ n) complexity applies only to fixed-k numerical evaluation using exponentiation by squaring. It does not apply to full symbolic expansion, since
deg⁡Pn(k)=4n+2.

Comment 4. Asymptotic growth.
We reformulated the asymptotic result more carefully. For fixed k with P2(k)≠0, the revised theorem states
limn→∞∣Pn(k)∣^1/n=∣ψ(k)∣
For integer k≥3, where Pn(k) and ψ(k) are positive, this reduces to
lim⁡n→∞Pn(k)^1/n=ψ(k)..

Comment 5. Scheduling interpretation.
We revised the scheduling section to correct the interpretation. The non-integer real root of ψ(k) is no longer interpreted as a minimum number of rooms. The minimum feasible number of rooms is now stated to be determined by the chromatic number:
kmin⁡=3.
We also clarified that ψ(k) measures the growth factor when one additional team is added while the number of rooms is fixed. The effect of increasing the number of rooms is now measured separately by ratios of the form
Pn(k2)/Pn(k1),k2>k1.

Comment 6. Mapping between the graph and the scheduling model.
We expanded the scheduling model by adding a precise vertex-to-session and edge-to-conflict mapping. The revised version explains that the two central vertices represent coordinator sessions in the two periods, while the 4n peripheral vertices represent two team-related sessions for each of the n teams in each of the two periods. Hence the model has exactly
2+4n=4n+2
sessions, matching the vertex count of Fn×P2. The edge constraints are also described explicitly.

We thank the reviewer again for the detailed comments. We believe that the revised manuscript is now more rigorous, clearer, and more precise in its mathematical derivation and scheduling interpretation.
Competing Interests: No competing interests. Close
Report a concern

Views

Reviewer Report 28 Apr 2026

Wafiq Hibi, Academic College of Sakhni, Sakhnin, Israel

Approved with Reservations

https://doi.org/10.5256/f1000research.195018.r476650

The manuscript studies the chromatic polynomial of the graph family F n × P 2 , where Fn denotes the friendship graph and P2 is the path on two vertices. The topic is relevant to algebraic and combinatorial graph theory, and the paper addresses a graph family that has received limited direct attention in the literature. The manuscript is generally well organized and the results are presented in a logical sequence.
General Evaluation:
The main contribution of the paper is the derivation of a recurrence relation and a closed-form expression for the chromatic polynomial of F n × P 2, together with structural properties, chromatic number analysis, asymptotic interpretation, and an illustrative scheduling application. The central formula appears mathematically consistent, and the numerical examples support the stated recurrence. The overall direction of the work is sound and potentially useful for researchers interested in graph products and chromatic invariants.

Strengths

The paper studies a nontrivial and specialized family of Cartesian product graphs.
The recurrence framework is elegant and gives a compact closed-form representation.
The structural results (order, size, chromatic number) are useful and coherent.
Numerical verification is included and supports the theoretical formulas.
The scheduling section provides a practical interpretation of the graph-coloring model.

Main Comments and Recommendations
1. Proof of the Transition Polynomial
The most important result is the derivation of the transition polynomial
ψ(k)= = k 4 − 8 k 3 + 26 k 2 − 41 k + 26
Although the result appears correct, the derivation is presented too briefly. The manuscript states that Inclusion–Exclusion is applied to eight constraints, but the key coefficients (26, 41, 26) are not derived transparently. For mathematical clarity and reproducibility, the authors should expand this section substantially.
Recommendation: Include either:

a complete combinatorial derivation, or
a structured case analysis, or
a supplementary appendix containing the full counting argument.

2. Recursive Independence Argument
The argument that the newly attached block contributes independently once the colors of the two central vertices are fixed is reasonable, but it would benefit from a more formal proof and clearer notation.
Recommendation: State the independence lemma more explicitly and clarify why no additional constraints arise from previous blocks.
3. Notation and Typography
Several expressions suffer from formatting inconsistencies (subscripts, spacing, symbols, repeated notation, typographical artifacts). This occasionally affects readability.
Recommendation: Carefully revise notation throughout the manuscript and ensure all formulas are typeset consistently.
4. Scheduling Application
The application is interesting as a motivating example, but it should be presented clearly as an illustrative use-case rather than a major applied breakthrough.
Recommendation: Shorten slightly or explicitly frame it as a demonstration of potential applicability.
Final Recommendation
This is a worthwhile mathematical contribution with a correct and interesting central result. However, the exposition of the main proof should be strengthened before final acceptance. After moderate revision focused on rigor, clarity, and presentation, the paper would be suitable for indexing.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

Yes

References

1. Hibi, W. (2022). Assembling Planer Graphs to Service the Coloring Number. Review of International Geographical Education Online, 12(1), 28-31.

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Mathematics Education and Applied Graph Theory in Teaching and Learning Contexts

CITE

Report a concern

Author Response 12 Jun 2026

Sarah M. Talab, Mathematics, Tikrit University, Tikrit, 34001, Iraq

12 Jun 2026

Author Response

We sincerely thank the reviewer for the careful reading of the manuscript and for the constructive comments. We appreciate the positive assessment of the topic, the recurrence framework, the structural ... Continue reading We sincerely thank the reviewer for the careful reading of the manuscript and for the constructive comments. We appreciate the positive assessment of the topic, the recurrence framework, the structural results, the numerical verification, and the scheduling interpretation. We have revised the manuscript to improve rigor, clarity, notation, and presentation.

Comment 1. Proof of the transition polynomial.
We agree that the derivation of
ψ(k)=k^4−8k^3+26k^2−41k+26\
needed to be expanded. In the revised manuscript, Section 3.2 now contains a structured combinatorial case analysis for the local extension count Ndiff(k). After fixing the colors of the two central vertices, we count the admissible colorings of the four new peripheral vertices and obtain
Ndiff(k)=2(k−2)^3+(k−2)(k−3)(k^2−5k+7)=ψ(k).
A summary table of the case analysis was also added.

Comment 2. Recursive independence argument.
We added Lemma 3.1.1 to formalize the conditional independence of the newly attached block Bn. The lemma clarifies that the new peripheral vertices have no edges to peripheral vertices of earlier blocks; hence, once the two central colors are fixed, all remaining constraints are local to Bn. This justifies the recurrence.

Comment 3. Notation and typography.
We revised the manuscript for notational consistency, subscripts, spacing, formula formatting, table presentation, and typographical artifacts. We also clarified that × denotes the Cartesian product throughout the paper.

Comment 4. Scheduling application.
We revised the scheduling section as an illustrative two-period model. The revised version includes explicit vertex-to-session and edge-to-conflict mappings, clarifies that χ(Fn×P2)=3 gives the minimum feasible number of rooms, and interprets ψ(k) correctly as the fixed-room team-growth factor rather than the effect of adding rooms. The assumptions and limitations of the model were also stated explicitly.

We thank the reviewer again for the helpful comments. We believe the revised version now provides a clearer derivation, a stronger recurrence argument, and a more precise interpretation of the scheduling application.
We sincerely thank the reviewer for the careful reading of the manuscript and for the constructive comments. We appreciate the positive assessment of the topic, the recurrence framework, the structural results, the numerical verification, and the scheduling interpretation. We have revised the manuscript to improve rigor, clarity, notation, and presentation.

Comment 1. Proof of the transition polynomial.
We agree that the derivation of
ψ(k)=k^4−8k^3+26k^2−41k+26\
needed to be expanded. In the revised manuscript, Section 3.2 now contains a structured combinatorial case analysis for the local extension count Ndiff(k). After fixing the colors of the two central vertices, we count the admissible colorings of the four new peripheral vertices and obtain
Ndiff(k)=2(k−2)^3+(k−2)(k−3)(k^2−5k+7)=ψ(k).
A summary table of the case analysis was also added.

Comment 2. Recursive independence argument.
We added Lemma 3.1.1 to formalize the conditional independence of the newly attached block Bn. The lemma clarifies that the new peripheral vertices have no edges to peripheral vertices of earlier blocks; hence, once the two central colors are fixed, all remaining constraints are local to Bn. This justifies the recurrence.

Comment 3. Notation and typography.
We revised the manuscript for notational consistency, subscripts, spacing, formula formatting, table presentation, and typographical artifacts. We also clarified that × denotes the Cartesian product throughout the paper.

Comment 4. Scheduling application.
We revised the scheduling section as an illustrative two-period model. The revised version includes explicit vertex-to-session and edge-to-conflict mappings, clarifies that χ(Fn×P2)=3 gives the minimum feasible number of rooms, and interprets ψ(k) correctly as the fixed-room team-growth factor rather than the effect of adding rooms. The assumptions and limitations of the model were also stated explicitly.

We thank the reviewer again for the helpful comments. We believe the revised version now provides a clearer derivation, a stronger recurrence argument, and a more precise interpretation of the scheduling application.
Competing Interests: No competing interests. Close
Report a concern
Respond or Comment

COMMENTS ON THIS REPORT

Author Response 12 Jun 2026

Sarah M. Talab, Mathematics, Tikrit University, Tikrit, 34001, Iraq

12 Jun 2026

Author Response

We sincerely thank the reviewer for the careful reading of the manuscript and for the constructive comments. We appreciate the positive assessment of the topic, the recurrence framework, the structural ... Continue reading We sincerely thank the reviewer for the careful reading of the manuscript and for the constructive comments. We appreciate the positive assessment of the topic, the recurrence framework, the structural results, the numerical verification, and the scheduling interpretation. We have revised the manuscript to improve rigor, clarity, notation, and presentation.

Comment 1. Proof of the transition polynomial.
We agree that the derivation of
ψ(k)=k^4−8k^3+26k^2−41k+26\
needed to be expanded. In the revised manuscript, Section 3.2 now contains a structured combinatorial case analysis for the local extension count Ndiff(k). After fixing the colors of the two central vertices, we count the admissible colorings of the four new peripheral vertices and obtain
Ndiff(k)=2(k−2)^3+(k−2)(k−3)(k^2−5k+7)=ψ(k).
A summary table of the case analysis was also added.

Comment 2. Recursive independence argument.
We added Lemma 3.1.1 to formalize the conditional independence of the newly attached block Bn. The lemma clarifies that the new peripheral vertices have no edges to peripheral vertices of earlier blocks; hence, once the two central colors are fixed, all remaining constraints are local to Bn. This justifies the recurrence.

Comment 3. Notation and typography.
We revised the manuscript for notational consistency, subscripts, spacing, formula formatting, table presentation, and typographical artifacts. We also clarified that × denotes the Cartesian product throughout the paper.

Comment 4. Scheduling application.
We revised the scheduling section as an illustrative two-period model. The revised version includes explicit vertex-to-session and edge-to-conflict mappings, clarifies that χ(Fn×P2)=3 gives the minimum feasible number of rooms, and interprets ψ(k) correctly as the fixed-room team-growth factor rather than the effect of adding rooms. The assumptions and limitations of the model were also stated explicitly.

We thank the reviewer again for the helpful comments. We believe the revised version now provides a clearer derivation, a stronger recurrence argument, and a more precise interpretation of the scheduling application.
We sincerely thank the reviewer for the careful reading of the manuscript and for the constructive comments. We appreciate the positive assessment of the topic, the recurrence framework, the structural results, the numerical verification, and the scheduling interpretation. We have revised the manuscript to improve rigor, clarity, notation, and presentation.

Comment 1. Proof of the transition polynomial.
We agree that the derivation of
ψ(k)=k^4−8k^3+26k^2−41k+26\
needed to be expanded. In the revised manuscript, Section 3.2 now contains a structured combinatorial case analysis for the local extension count Ndiff(k). After fixing the colors of the two central vertices, we count the admissible colorings of the four new peripheral vertices and obtain
Ndiff(k)=2(k−2)^3+(k−2)(k−3)(k^2−5k+7)=ψ(k).
A summary table of the case analysis was also added.

Comment 2. Recursive independence argument.
We added Lemma 3.1.1 to formalize the conditional independence of the newly attached block Bn. The lemma clarifies that the new peripheral vertices have no edges to peripheral vertices of earlier blocks; hence, once the two central colors are fixed, all remaining constraints are local to Bn. This justifies the recurrence.

Comment 3. Notation and typography.
We revised the manuscript for notational consistency, subscripts, spacing, formula formatting, table presentation, and typographical artifacts. We also clarified that × denotes the Cartesian product throughout the paper.

Comment 4. Scheduling application.
We revised the scheduling section as an illustrative two-period model. The revised version includes explicit vertex-to-session and edge-to-conflict mappings, clarifies that χ(Fn×P2)=3 gives the minimum feasible number of rooms, and interprets ψ(k) correctly as the fixed-room team-growth factor rather than the effect of adding rooms. The assumptions and limitations of the model were also stated explicitly.

We thank the reviewer again for the helpful comments. We believe the revised version now provides a clearer derivation, a stronger recurrence argument, and a more precise interpretation of the scheduling application.
Competing Interests: No competing interests. Close
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VERSION 2 PUBLISHED 04 Mar 2026

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Version 2 (revision) 12 Jun 26
Version 1 04 Mar 26	read	read

Wafiq Hibi, Academic College of Sakhni, Sakhnin, Israel
Siti Amiroch, Universitas Islam Darul ‘ulum, Lamongan, Indonesia

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Reviewer Report

14 Views

11 May 2026 | for Version 1

Siti Amiroch, Universitas Islam Darul ‘ulum, Lamongan, Indonesia

14 Views Cite this report Responses(1)

Approved With Reservations

Reviewer Report

The manuscript studies the chromatic polynomial of the graph family FnxP2

, where Fn

denotes the friendship graph and P2

is non-planar for n ≥ 3 is not sufficiently justified. The manuscript states that the graph contains a K3,3

should be formulated more carefully. For general real or complex values of k, the mathematically safer formulation should involve Pn(k)1n

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Partly
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests

No competing interests were disclosed.

Respond to this report

Responses (1)

Author Response

12 Jun 2026

Sarah M. Talab, Mathematics, Tikrit University, Tikrit, 34001, Iraq

We sincerely thank the reviewer for the careful evaluation and constructive comments. We appreciate the reviewer’s recognition of the relevance of the topic, the recursive approach, the chromatic number result, the numerical verification, and the potential scheduling interpretation. We have revised the manuscript substantially to address all points raised.

Comment 1. Derivation of the transition polynomial.
We agree that the derivation of ψ(k) required more detail. In the revised manuscript, Section 3.2 now includes a structured combinatorial case analysis for the local extension count Ndiff(k). After fixing the colors of the two central vertices, we count the admissible colorings of the four new peripheral vertices and obtain
Ndiff(k)=2(k−2)^3+(k−2)(k−3)(k^2−5k+7)=ψ(k).This gives
ψ(k)=k^4−8k^3+26k^2−41k+26.
A summary table of the enumeration has also been added.

Comment 2. Non-planarity claim.
We agree that the previous non-planarity statement required a rigorous proof. To avoid unsupported claims outside the main focus of the paper, we removed this assertion from the revised manuscript. The structural analysis now focuses on the order, size, connectivity, and degree sequence, which are directly used in the chromatic analysis.

Comment 3. Computational complexity.
We revised the complexity statement. The manuscript now states explicitly that the O(log⁡ n) complexity applies only to fixed-k numerical evaluation using exponentiation by squaring. It does not apply to full symbolic expansion, since
deg⁡Pn(k)=4n+2.

Comment 4. Asymptotic growth.
We reformulated the asymptotic result more carefully. For fixed k with P2(k)≠0, the revised theorem states
limn→∞∣Pn(k)∣^1/n=∣ψ(k)∣
For integer k≥3, where Pn(k) and ψ(k) are positive, this reduces to
lim⁡n→∞Pn(k)^1/n=ψ(k)..

Comment 5. Scheduling interpretation.
We revised the scheduling section to correct the interpretation. The non-integer real root of ψ(k) is no longer interpreted as a minimum number of rooms. The minimum feasible number of rooms is now stated to be determined by the chromatic number:
kmin⁡=3.
We also clarified that ψ(k) measures the growth factor when one additional team is added while the number of rooms is fixed. The effect of increasing the number of rooms is now measured separately by ratios of the form
Pn(k2)/Pn(k1),k2>k1.

Comment 6. Mapping between the graph and the scheduling model.
We expanded the scheduling model by adding a precise vertex-to-session and edge-to-conflict mapping. The revised version explains that the two central vertices represent coordinator sessions in the two periods, while the 4n peripheral vertices represent two team-related sessions for each of the n teams in each of the two periods. Hence the model has exactly
2+4n=4n+2
sessions, matching the vertex count of Fn×P2. The edge constraints are also described explicitly.

We thank the reviewer again for the detailed comments. We believe that the revised manuscript is now more rigorous, clearer, and more precise in its mathematical derivation and scheduling interpretation.

View more View less

Competing Interests

No competing interests.

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Reviewer Report

14 Views

28 Apr 2026 | for Version 1

Wafiq Hibi, Academic College of Sakhni, Sakhnin, Israel

14 Views Cite this report Responses(1)

Approved With Reservations

The paper studies a nontrivial and specialized family of Cartesian product graphs.
The recurrence framework is elegant and gives a compact closed-form representation.
The structural results (order, size, chromatic number) are useful and coherent.
Numerical verification is included and supports the theoretical formulas.
The scheduling section provides a practical interpretation of the graph-coloring model.

a complete combinatorial derivation, or
a structured case analysis, or
a supplementary appendix containing the full counting argument.

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

Yes

References

1. Hibi, W. (2022). Assembling Planer Graphs to Service the Coloring Number. Review of International Geographical Education Online, 12(1), 28-31.

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Mathematics Education and Applied Graph Theory in Teaching and Learning Contexts

Respond to this report

Responses (1)

Author Response

12 Jun 2026

Sarah M. Talab, Mathematics, Tikrit University, Tikrit, 34001, Iraq

We sincerely thank the reviewer for the careful reading of the manuscript and for the constructive comments. We appreciate the positive assessment of the topic, the recurrence framework, the structural results, the numerical verification, and the scheduling interpretation. We have revised the manuscript to improve rigor, clarity, notation, and presentation.

Comment 1. Proof of the transition polynomial.
We agree that the derivation of
ψ(k)=k^4−8k^3+26k^2−41k+26\
needed to be expanded. In the revised manuscript, Section 3.2 now contains a structured combinatorial case analysis for the local extension count Ndiff(k). After fixing the colors of the two central vertices, we count the admissible colorings of the four new peripheral vertices and obtain
Ndiff(k)=2(k−2)^3+(k−2)(k−3)(k^2−5k+7)=ψ(k).
A summary table of the case analysis was also added.

Comment 2. Recursive independence argument.
We added Lemma 3.1.1 to formalize the conditional independence of the newly attached block Bn. The lemma clarifies that the new peripheral vertices have no edges to peripheral vertices of earlier blocks; hence, once the two central colors are fixed, all remaining constraints are local to Bn. This justifies the recurrence.

Comment 3. Notation and typography.
We revised the manuscript for notational consistency, subscripts, spacing, formula formatting, table presentation, and typographical artifacts. We also clarified that × denotes the Cartesian product throughout the paper.

Comment 4. Scheduling application.
We revised the scheduling section as an illustrative two-period model. The revised version includes explicit vertex-to-session and edge-to-conflict mappings, clarifies that χ(Fn×P2)=3 gives the minimum feasible number of rooms, and interprets ψ(k) correctly as the fixed-room team-growth factor rather than the effect of adding rooms. The assumptions and limitations of the model were also stated explicitly.

We thank the reviewer again for the helpful comments. We believe the revised version now provides a clearer derivation, a stronger recurrence argument, and a more precise interpretation of the scheduling application.

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Competing Interests

No competing interests.

Alongside their report, reviewers assign a status to the article:

Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

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Chromatic Polynomials of Fn×P2 Graphs: Algebraic Analysis and Scheduling Applications

Abstract

Background

Methods

Results

Conclusions

Keywords

Revised Amendments from Version 1

1. Introduction

2. Preliminaries

Definition 2.1

Definition 2.2

Definition 2.3

Definition 2.4:

Figure 1. Friendship graphs Fn for n=2,3,4,5.

Figure 2. Structure of the Cartesian Product Fn×P2 for n=2,3,4,5.

3. Methods

3.1 Analytical framework and structural decomposition

Lemma 3.1.1

Combinatorial derivation of the transition polynomial ψ(k)

3.2.

Table 1. Case analysis for the local extension count Ndiff(k)=ψ(k) .

Derivation of the recurrence and closed-form expression

3.3.

4. Results

4.1 Structural analysis of Fn×P2

Theorem 4.1.1:

Proof:

Remark 4.1.2:

Table 2. Edge classification in Fn×P2 .

4.2 Chromatic polynomial analysis

Theorem 4.2.1:

Proof:

Theorem 4.2.2:

Proof:

Corollary 4.2.3

Proposition 4.2.4

Proof:

4.3 Algebraic and asymptotic analysis

Theorem 4.3.1

Proof:

Theorem 4.3.2

Proof:

Remark 4.3.3

Corollary 4.3.4

Proof:

Remark 4.3.5

4.4 Numerical validation

Table 3. Numerical values of Pn(k) for n=2,3,4,and5 , along with ψ(k) at integer values k≥3 .

4.5 Illustrative application: a two-period scheduling model

Table 4. Scheduling with the minimum number of rooms (k=3) .

Table 5. Scalability for fixed room availability.

Table 6. Effect of increasing the number of rooms for fixed n=5 .

Table 7. From Graph Elements to Scheduling Interpretation.

5. Discussion

Theoretical contribution

5.1.

Structural, computational, and scheduling implications

5.2

Limitations and future directions

5.3

Ethical considerations

Use of AI-assisted technology

Data availability

Reporting guidelines

Acknowledgements

References

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Figure 1. Friendship graphs $F_{n}$ for $n = 2, 3, 4, 5 .$

Figure 2. Structure of the Cartesian Product $F_{n} \times P_{2}$ for $n = 2, 3, 4, 5 .$

Combinatorial derivation of the transition polynomial $ψ (k)$

Table 1. Case analysis for the local extension count $N_{diff} (k) = ψ (k)$ .

4.1 Structural analysis of $F_{n} \times P_{2}$

Table 2. Edge classification in $F_{n} \times P_{2}$ .

Table 3. Numerical values of $P_{n} (k)$ for $n = 2, 3, 4, and 5$ , along with $ψ (k)$ at integer values $k \geq 3$ .

Table 4. Scheduling with the minimum number of rooms $(k = 3)$ .

Table 6. Effect of increasing the number of rooms for fixed $n = 5$ .