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

Sarah M. Talab; Nabeel E. Arif

doi:10.12688/f1000research.176896.1

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Research Article

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

[version 1; peer review: 2 approved with reservations]

Sarah M. Talab ¹, Nabeel E. Arif¹

PUBLISHED 04 Mar 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, bridging pure mathematics and practical applications. While extensive results exist for paths and cycles, the Cartesian product F n × P 2 remains largely unexplored despite its layered constraint structure, presenting a clear gap in the literature.

Methods

We employ combinatorial decomposition and recursive block construction, applying the inclusion–exclusion principle to the eight edge constraints within each recursive unit. This analytical approach enables the derivation of the chromatic transition polynomial ψ ( k ) , which governs the recurrence relations and closed-form expressions.

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 . The chromatic number is proven to be χ = 3 , with real roots of ψ ( k ) located within [2,3]. Numerical validation confirms both recurrence and closed-form formulas, while asymptotic analysis shows the exponential growth of P n ( k ) is governed by ψ ( k ) , as lim n → ∞ [ P n ( k ) ] 1 n = | ψ ( k ) | .

Conclusions

This research provides a comprehensive algebraic characterization of the chromatic polynomial for F n × P 2 , deriving its recurrence relation and closed-form expression. Building on this foundation, we develop a novel two-period conference scheduling model where the chromatic polynomial serves as a quantitative tool to compute all conflict-free room allocations. This work demonstrates directly how structural graph theory can inform practical resource allocation systems, transforming an abstract invariant into a concrete decision-support tool.

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 1; peer review: 2 approved with reservations]. F1000Research 2026, 15:351 (https://doi.org/10.12688/f1000research.176896.1) 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)

1. Introduction

Chromatic polynomials are considered a fundamental tool in algebraic graph theory. Initially introduced by Birkhoff (1912) in his attempt to prove the four-color conjecture, their development was profoundly advanced by Whitney’s (1932) deletion-contraction recurrence and Read’s (1968) systematic studies, culminating in a comprehensive review by Tutte and Read (1988).

The chromatic polynomial $P (G, k)$ , which counts the proper $k$ -colorings of a graph $G$ , also bears significant practical importance. It serves as a critical tool in diverse applied fields, including task scheduling,¹ data analysis,² network design,³ theoretical chemistry,⁴ and statistical physics.⁵ However, computing the chromatic polynomial remains a challenging problem, particularly for graphs constructed from Cartesian products $(G \times H)$ , where determining a general formula relating $P (G \times H, k)$ to its factor polynomials is $NP$ -hard. This challenge has encouraged the derivation of closed-form expressions for specific graph classes.

The properties of Cartesian products, including their coloring characteristics, have provided a solid theoretical basis for studying composite graphs.⁶ The effect of these graph operations on coloring and dominance has been widely investigated as a key to understanding the structure of composite graphs.⁷ Notably, recent studies have successfully derived chromatic polynomials for other composite structures, such as the Triangular Snake and the n-Centipede graphs using structural recursion.⁸

While significant research has focused on Cartesian products of basic graphs, such as paths ( $P_{n}$ ) and cycles( $C_{n}$ ),^9–12 the structure $F_{n} \times P_{2}$ —formed by the Cartesian product of a friendship graph and a 2-path—has received little attention. Its unique composition, which interlaces local triangular clusters with a linear, two-layer framework, presents a compelling subject for algebraic graph theory.

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

• Deriving its recurrence relation and a closed-form expression for its chromatic polynomial.
• Establishing its chromatic number, analyzing the root distribution of its transition polynomial, and determining its asymptotic growth rate.
• Validating the theoretical results through numerical computation.
• Demonstrating its practical utility via a novel application to a two-period conference scheduling problem.

This practical approach aligns with recent results^1,13 that confirm the utility of graph coloring in modeling scheduling constraints and resolving resource conflicts, thereby reinforcing the practical relevance of our findings.

2. Preliminaries

Definition 2.1

¹⁴: A friendship graph, denotes $F_{n}$ for $n \geq 2$ , and it’s defined as the union of $n$ copies of cycles $(C_{3})$ with a common vertex (the center). Formally:

• Vertex set: $V (F_{n}) = {v_{0}} \cup {u_{i}, w_{i}}_{i = 1}^{n}$ where $v_{0}$ is the center vertex.
• Edge set: $E (F_{n}) = {(v_{0}, u_{i}), (v_{0}, w_{i}), (u_{i}, w_{i})}_{i = 1}^{n}$ .
• Order: $| V (F_{n}) | = 2 n + 1$ .
• Size: $| E (F_{n}) | = 3 n$ .
Figure 1 (Friendship graphs F_n):

Definition 2.2

⁶: The Cartesian product of two graphs $G$ and $H$ denotes $(G \times H)$ and it’s known as a graph in which each vertex is an ordered pair $(u, w)$ where $u \in V (G) and w \in V (H)$ , creating an edge between two vertices $(u_{1}, w_{1})$ and $(u_{2}, w_{2})$ .

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 $G$ .

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 construction. For $n \geq 3$ , $G_{n}$ is obtained from $G_{n - 1}$ by attaching a new block $B_{n}$ .This block contains four new vertices ${u_{n}^{A}, w_{n}^{A}, u_{n}^{B}, w_{n}^{B}}$ and eight edges that form two new triangles (one in each layer) along with their vertical connections. Crucially, $B_{n}$ attaches only to the two central vertices $v_{0}^{A}$ and $v_{0}^{B}$ of $G_{n - 1}$ . This localized, exclusive attachment is the key to isolating the chromatic contribution of each step.

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

The proper coloring of the new block $B_{n}$ depends solely on the colors assigned to its two attachment points, $v_{0}^{A}$ and $v_{0}^{B}$ , which must be distinct in any proper coloring of the base graph $G_{n - 1}$ . We compute the number of proper $k$ -colorings of $B_{n}$ under this condition, denoted $N_{diff} (k)$ , using the inclusion-exclusion principle applied to its eight edge constraints.

Let $S$ be the set of all colorings of the four new vertices without constraints, so $| S | = k^{4}$ . For an edge $e_{i}$ , let $M_{i}$ be the set of colorings where its endpoints share the same color. Then:

N_{diff} (k) = | S | - \sum_{i = 1}^{8} | M_{i} | + \sum_{1 \leq i < j \leq 8} | M_{i} \cap M_{j} | - \sum_{1 \leq i < j < t \leq 8} | M_{i} \cap M_{j} \cap M_{t} | + \dots + {(- 1)}^{8} | M_{1} \cap M_{2} \cap \dots \cap M_{8} |

Coefficient Analysis:

$- 8 k^{3}$ : Each of the 8 edges defines a single constraint $| M_{i} | = k^{3}$ , because fixing the color of one endpoint (or satisfying the equality) leaves 3 vertices free.

$+ 26 k^{2}$ : There are 26 compatible edge pairs (out of 28) that can be satisfied simultaneously without color conflicts under the distinct‑central‑colors condition, each giving $| M_{i} \cap M_{j} | = k^{2}$ .

$- 41 k$ : Analysis of compatible edge triples yields 41 configurations with $| M_{i} \cap M_{j} \cap M_{t} | = k$ .

$+ 26$ : The full intersection of all eight constraints corresponds to 26 distinct colorings consistent with the distinct central colors.

This combinatorial enumeration yields the chromatic transition polynomial:

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

This result is algebraically verified by exact polynomial division: $ψ (k) = \frac{P_{3} (k)}{P_{2} (k)}$ for all $k \geq 3 .$

Remark 3.2.1

(Methodological Soundness)

The polynomial $ψ (k)$ is validated through dual independent approaches:

The combinatorial inclusion–exclusion derivation provides structural insight into the constraint system, while the exact polynomial division $(ψ (k) = \frac{P_{3} (k)}{P_{2} (k)})$ offers algebraic confirmation, ensuring mathematical rigor.

3.3 Derivation of the recurrence and closed-form expression

The structural isolation of $B_{n}$ implies that any proper coloring of $G_{n}$ can be obtained by independently choosing:

(i) a proper coloring of $G_{n - 1}$ and
(ii) a proper coloring of $B_{n}$ consistent with the colors of the central vertices in $G_{n - 1}$ . Consequently, the chromatic polynomials satisfy the fundamental recurrence relation for $n \geq 3$ :

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

Applying mathematical induction to this recurrence, with $P_{2} (k)$ as the verified base case, provides the closed-form expression for all $n \geq 2$ :

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

3.4 Additional analytical and numerical validation

• Chromatic Number: Combinatorial reasoning (the presence of disjoint triangles and an explicit constructive 3-coloring) establishes $χ = 3$ .
• Root Analysis: Using calculus (evaluation of $ψ (k)$ and its derivative, the Intermediate Value Theorem), we prove the real roots of $ψ (k)$ lie within the interval $[2, 3]$ .
• Asymptotic Growth Rate: The closed-form expression directly implies the exponential growth rate: $lim_{n \to \infty} {[P_{n} (k)]}^{\frac{1}{n}} = | ψ (k) |$ .
• Numerical Validation: All derived formulas are validated for integer values $k \geq 3$ using Wolfram Mathematica confirming the consistency of the recurrence and closed-form expression with directly computed values of $P_{n} (k)$ .

4. Results

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

Theorem 4.1.1:

(Structural Properties and Chromatic Implications)

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: Every $F_{n}$ has $(2 n + 1)$ vertices. The product with $P_{2}$ resulting in $2 \times (2 n + 1) = 4 n + 2$ vertices. This layering allows a recursive coloring process.
2. Edge count consists of:
- • Internal edges: $2 \times 3 n = 6 n$ ,
- • Vertical edges: $2 n$ (connecting peripherals) + $1$ (connecting centers) = $2 n + 1$ ,
We get: $6 n + (2 n + 1) = 8 n + 1$ .
3. Degree analysis:
- • Central Vertices ( $2$ vertices): Every center is connected to:
  - $2 n$ peripheral vertices in its layer.
  - 1 center vertex in the opposite layer.

deg = 2 n + 1 .

• Peripheral Vertices ( $4 n$ vertices): Every peripheral vertex is connected to:
- 1 center vertex in its layer.
- 1 partner vertex in the same triangle.
- 1 opposite vertex in the opposite layer.
  $deg = 3 .$

Thus, the degree sequence is ${{(2 n + 1)}^{(2)}, 3^{(4 n)}}$ . ∎

Chromatic Significance:

This structure includes a hierarchical constraint system:

• Central vertices act as chromatic regulators, helping with color separation in both layers.
• Peripheral vertices form recursive units, which have local coloring constraints.
• Horizontal edges uphold proper coloring across time periods.
• Vertical edges prevent color reuse across sequential periods.

Theorem 4.1.2:

(Edge classification and Distribution)

The edge classification derives directly from the vertex degree analysis in Theorem 4.1.1 ( Table 1).

Proof:

The three edge types and their counts are determined as follows:

1. Central Edge: Exactly one edge connects the two central vertices, each of degree ( $2 n + 1)$ .
2. Center-Peripheral Edges: Each central vertex is adjacent to $2 n$ peripheral vertices in its own layer, yielding $2 \times 2 n = 4 n$ edges of this type.
3. Uniform Peripheral Edges: This set comprises:
- • The base edges of the n triangles in each layer: $2 \times n = 2 n$ edges.
- • The vertical edges connecting corresponding peripheral vertices across layers: $2 n$ edges.

Their total is $2 n + 2 n = 4 n$ edges.

This completes the classification. ∎

Proposition 4.1.3:

(Graph Properties)

The graph $F_{n} \times P_{2}$ is:

1. Connected, for $n \geq 2$ .
2. Non-planar for $n \geq 3$ .

Proof:

1. Since $F_{n}$ is connected, the Cartesian product with P₂ adds a second layer that is linked to the first through vertical edges between corresponding vertices. These vertical connections ensure that the two layers are joined, so the resulting graph F_n × P₂ remains connected.
2. For $n \geq 3$ , it contains $Κ_{3, 3}$ minor; therefore, by Kuratowski’s theorem, it is non-planar. ∎

Lemma 4.1.4:

(Isolation Lemma for the Block $B_{n}$ )

In the recursive construction of $G_{n} = F_{n} \times P_{2}$ ( Section 3.1), the coloring of the newly attached block $B_{n}$ is conditionally independent of the coloring of the subgraph $G_{n - 1}$ . Formally, given any fixed pair of distinct colors assigned to the central vertices $v_{0}^{A}$ and $v_{0}^{B}$ , the number of valid extensions to color the four vertices of $B_{n}$ is a well-defined function $N_{diff} (k)$ that depends only on the number of colors $k$ .

Proof:

The block $B_{n}$ is adjacent to $G_{n - 1}$ only at the vertices $v_{0}^{A}$ and $v_{0}^{B}$ . No edge connects $B_{n}$ to any other vertex of $G_{n - 1}$ . Therefore, once colors for $v_{0}^{A}$ and $v_{0}^{B}$ are fixed, the coloring constraints are entirely contained within $B_{n}$ , making the extension count independent of the rest of $G_{n - 1}$ . The symmetry of the coloring problem under permutations of the color set ensures that this count depends only on $k$ . ∎

Table 1. Edge classification in $F_{n} \times P_{2}$ .

Edge type	Description	Degree of Endpoints	Count
Central Edge	Connects the two central vertices	$(2 n + 1, 2 n + 1)$	1
Center-Peripheral Edges	Connects central to peripheral vertices	$(2 n + 1, 3)$	$4 n$
Uniform Peripheral Edges	Connects degree- $3$ vertices	$(3, 3)$	$4 n$

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 governed by the chromatic transition polynomial $ψ (k)$ :

P_{n} (k) = ψ (k) P_{n - 1} (k),

where

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

Proof:

The factorization $P_{n} (k) = N_{diff} (k) P_{n - 1} (k)$ follows from the conditional independence in the recursive construction (Lemma 4.1.4). The equality $N_{diff} (k) = ψ (k)$ is the result of the combinatorial enumeration detailed in Section 3.2. ∎

Theorem 4.2.2:

(Closed-Form Expression)

For $n \geq 2$ , the chromatic polynomial of $F_{n} \times P_{2}$ is given by the closed-form expression:

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

Proof:

By using mathematical induction on $n$ :

Base case $(n = 2)$ : Trivially, $P_{2} (k) = {[ψ (k)]}^{0} P_{2} (k)$ .

Suppose the formula holds for $n - 1$ , i.e.,

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

using the recurrence relation

P_{n} (k) = ψ (k) P_{n - 1} (k),

we substitute the inductive hypothesis:

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

∎

Corollary 4.2.3

(Computational Efficiency)

The closed-form expression $P_{n} (k) = {[ψ (k)]}^{n - 2} P_{2} (k)$ . reduces the time complexity of computing $P_{n} (k)$ from exponential (via the deletion–contraction algorithm) to $O (log n)$ using exponentiation by squaring, providing a significant computational advantage for large $n$ .

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 triangles (copies of $C_{3}$ ), at least 3 colors are required, establishing the lower bound $χ \geq 3$ . To show that 3 colors suffice, we construct an explicit proper 3-coloring $c : V \to {1, 2, 3}$ . Color the two center vertices as $c (v_{0}^{A}) = 1$ and $c (v_{0}^{B}) = 2$ . For each triangle $i (i = 1, \dots, n)$ , assign colors to the peripheral vertices as follows: $c (u_{i}^{A}) = 2, c (w_{i}^{A}) = 3$ , $c (u_{i}^{B}) = 3, c (w_{i}^{B}) = 1$ . One may verify that all edges within triangles, between centers, and vertical edges between layers receive distinct colors at their endpoints. Thus, $c$ is a proper 3-coloring, proving the upper bound $χ \leq 3$ . Therefore, $χ = 3$ .∎

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, both in the interval $[2, 3] .$

Proof:

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

The derivative $ψ^{'} (k) = 4 k^{3} - 24 k^{2} + 52 k - 41$ satisfies $ψ^{'} (2) = - 1 < 0$ , $ψ^{'} (2.5) = 1.5 > 0$ .

Existence: Since $ψ^{'} (2) < 0, ψ$ is decreasing at $k = 2$ , so $ψ (2 + ε) < 0$ for small $ε > 0$ . With $ψ (2.5) > 0$ and $ψ$ continuous, the Intermediate Value Theorem gives a root in $(2,2.5)$ .

Uniqueness: The second derivative $ψ^{″} (k) = 12 k^{2} - 48 k + 52$ has discriminant $- 192 < 0$ and positive leading coefficient, so $ψ^{″} > 0$ everywhere. Thus $ψ^{'}$ is strictly increasing and changes sign exactly once in $(2,2.5),$ giving at most one root. Combined with existence, the root is unique.

Hence $ψ$ has roots at $k = 2$ and in $(2,2.5)$ . By the Fundamental Theorem of Algebra, the remaining two roots are complex conjugates. ∎

Theorem 4.3.2

(Exponential Growth Rate)

For all $k$ such that $ψ (k) \neq 0$ and $P_{2} (k) \neq 0$ , we have

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

Proof:

From the closed-form expression in Theorem 4.2.2,

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

Write $ψ (k)$ in polar form:

ψ (k) = | ψ (k) | e^{iθ}, θ = arg (ψ (k)) .

Then:

{[P_{n} (k)]}^{\frac{1}{n}} = {| {[ψ (k)]}^{n - 2} P_{2} (k) |}^{\frac{1}{n}} = {| ψ (k) |}^{1 - \frac{2}{n}} . {| P_{2} (k) |}^{\frac{1}{n}} \cdot e^{iθ (1 - \frac{2}{n})}

Taking limits:

• $lim_{n \to \infty} {| ψ (k) |}^{1 - \frac{2}{n}} = | ψ (k) |$ ,

because $| ψ (k) |$ is constant and $1 - \frac{2}{n} \to 1$ .

• $lim_{n \to \infty} {[P_{2} (k)]}^{\frac{1}{n}} = 1$ ,

since $P_{2} (k)$ is a non‑zero constant polynomial, and for any constant $c > 0$ , $c^{\frac{1}{n}} \to 1$ .

• $| e^{iθ (1 - \frac{2}{n})} | = 1$ for all $n$ ,

since $| e^{iϕ} | = 1$ for every real $ϕ$

(here $ϕ = θ (1 - \frac{2}{n})$ ).

Therefore:

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

∎

Note: For integer $k \geq 3$ , Table 2 shows $ψ (k) > 0$ , hence in that case $| ψ (k) | = ψ (k)$ .

Corollary 4.3.3

(Ratio Convergence of Chromatic Polynomials)

For all $k$ such that $ψ (k) \neq 0$ , $lim_{n \to \infty} \frac{P_{n} (k)}{P_{n - 1} (k)} = ψ (k)$ .

Proof:

This follows immediately from the recurrence relation

$P_{n} (k) = ψ (k) P_{n - 1} (k)$ in Theorem 4.2.1.

For $n \geq 3$ , we have $\frac{P_{n} (k)}{P_{n - 1} (k)} = ψ (k)$ ,

so the limit as $n \to \infty$ is trivially $ψ (k)$ . ∎

Table 2. Numerical values of $P_{n} (k)$ for $= 2, 3, 4, and 5$ , 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

Interpretation. The value $| ψ (k) |$ serves as the exponential growth constant for the sequence ${P_{n} (k)}$ . This establishes $ψ (k)$ as the fundamental scaling factor governing the asymptotic expansion of chromatic polynomials. For large $n$ , $P_{n} (k)$ scales approximately as ${| ψ (k) |}^{n}$ , indicating that each increment in n multiplies the number of proper colorings by approximately $| ψ (k) |$ .

Remark 4.3.4

(Asymptotic Behavior and Root Distribution)

1. Theorem 4.3.2 establishes $| ψ (k) |$ as the exponential growth constant for the sequence $P_{n} (k)$ . For $k \geq 3, ψ (k)$ itself serves this role.
2. The roots of $P_{n} (k)$ comprise:
- • The fixed roots of $P_{2} (k)$ , and
- • The roots of $ψ (k)$ , where each root of $ψ (k)$ has an algebraic multiplicity of $n - 2$ .

As $n \to \infty$ , the roots of $ψ (k)$ dominate the overall distribution, acting as accumulation points in the complex plane.

4.4 Numerical validation

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

Extensive verification for integer values $k \geq 3$ showed perfect agreement among three independent approaches:

1. Direct computation of $P_{n} (k)$ ,
2. Evaluation using the recurrence relation,
3. Evaluation using the closed-form expression.

This numerical validation confirms the consistency of our theoretical results, as summarized in Table 2.

Specific calculations that demonstrate the application of the derived formulas are as follows:

• Theorem 4.2.1: $P_{3} (3) = ψ (3) . P_{2} (3) = 2 . 24 = 48$ .
• Theorem 4.2.2: $P_{5} (6) = {[ψ (6)]}^{3} P_{2} (6)$ $= 284^{3} . 2419680 = 55425925662720$ .
• Exponential growth: ${[P_{5} (4)]}^{\frac{1}{5}} = {(61843584)}^{\frac{1}{5}} \approx 22.000 \approx ψ (4)$ .

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 offer conclusive verification of Theorem 4.2.1 and Theorem 4.2.2 for the checked values. Furthermore, the convergence of ${[P_{n} (k)]}^{\frac{1}{n}}$ to $ψ (k)$ , numerically validates the exponential growth behavior established in Theorem 4.3.2.

4.5 Application: A chromatic model for hierarchical conference scheduling

The chromatic polynomial $P_{n} (k)$ of the graph $G = F_{n} \times P_{2}$ is used to model a constrained two-period conference scheduling system.

4.5.1 Model Specification: A Two-Period Conference

The conference structure is modeled by the friendship graph $F_{n}$ , where:

• Central vertex: Refers to the conference coordinator.
• Triangles: Refers to the research teams and their relationship to the coordinator.

Each team must complete two different tasks:

a. Project presentation.
b. Brainstorming and discussion.

These tasks are scheduled across two consecutive periods (e.g., morning and afternoon sessions). The system specifications include:

• Participants: One coordinator and $n$ independent teams.
• Sessions: Two time periods with two task types per team.
• Resources: $k$ identical meeting rooms.
• Objective: Assign rooms to all sessions while satisfying scheduling constraints.

4.5.2 Graph-Theoretic Representation

The graph $G = F_{n} \times P_{2}$ yield all scheduling constraints by its vertex and edge structure:

• Vertices: Refer to all sessions (coordinator and teams across both periods)
• Edges: Represent conflicts and constraints:
- - Horizontal edges: Prevent simultaneous room usage for related sessions.
- - Vertical edges: Prevent room reuse by the same team across consecutive periods.

4.5.3 Chromatic Polynomial as Scheduling Tool

The Chromatic polynomial $P_{n} (k)$ computes all valid room allocation schedules that satisfy the following critical conditions:

1. Conflict avoidance: No two conflicting sessions share the same room.
2. Temporal separation: No room reuse for the same team across consecutive periods.
3. Constraint compliance: All session-specific scheduling constraints are observed.

4.5.4 Analytical Planning Insights

The closed-form expression $P_{n} (k) = {[ψ (k)]}^{n - 2} P_{2} (k)$ offers important insights for conference planning:

• Feasibility Threshold: The real root of $ψ (k)$ at $k_{\min} \approx 2.5$ refers to the theoretical minimum rooms required, but the chromatic number $χ (F_{n} \times P_{2}) = 3$ shows the practical minimum, meaning that $3$ rooms are enough for any conference size.
• Scalability Analysis: The growth rate determined by $ψ (k)$ predicts how room demand scales with the increase in teams, establishing the fundamental mathematical relationship between conference size and resource requirements.
• Flexibility Quantification: The chromatic polynomial value $P_{n} (k)$ directly measures the theoretical flexibility space accessible to planners, with higher values indicating greater resilience against scheduling constraints.

These theoretical insights define the basic boundaries with possibilities of the scheduling system, which we will quantitatively check in the following section.

4.5.5 Quantitative Performance Analysis

Based on the theoretical framework, we conduct a performance analysis of the conference scheduling model using the calculated values of $P_{n} (k)$ . This translation turns abstract graph-theoretic concepts into concrete planning metrics. Our analysis focuses on three key performance measures derived from the chromatic model: (1) the practical feasibility threshold, validated by the chromatic number $χ = 3$ ; (2) the measured scheduling flexibility, quantified directly by the chromatic polynomial $P_{n} (k)$ ; and (3) the observed scalability indicator, defined by the growth rate of $P_{n} (k)$ with respect to $n$ .

Scheduling with the Minimum Required Rooms ( $k = 3$ )

The operational feasibility of the theoretical chromatic number is demonstrated numerically. For a growing number of teams $n$ , the number of valid conflict-free schedules $P_{n} (3)$ exhibits exact exponential growth, doubling with each added team: $P_{n} (3) = 2 . P_{n - 1} (3)$ . The specific values for conferences of different scales are compiled in Table 3, confirming that any conference with $n \geq 2$ teams can be scheduled with only three rooms.

Table 3. Conference scheduling with minimum rooms.

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

Quantifying the Impact of Additional Resources

To evaluate the gains from increased resources, we compare the scheduling flexibility $P_{n} (k)$ for $k = 3, 4, 5$ . The results, detailed in Table 4, reveal the principle of exponential resource leverage. A single additional room (moving from $k = 3$ to $k = 4$ ) increases the number of valid schedules for $5$ teams from $192$ to over $61.8$ million. This represents a gain factor of approximately $22$ , which equals $ψ (4)$ . Similarly, providing five rooms yields a gain factor of approximately 96, equaling $ψ (5)$ . These values were computed using the closed-form expression $P_{n} (k) = {[ψ (k)]}^{n - 2} P_{2} (k)$ , where $ψ (4) = 22$ and $ψ (5) = 96$ .

Table 4. Impact of room availability on scheduling flexibility.

Scenario	$k$	$P_{4} (k)$	$P_{5} (k)$	Gain $(P_{5} (k) vs P_{4} (k))$
Minimal Rooms	3	96	192	$\times 2$
Added Flexibility	4	2,811,072	61,843,584	$\approx$ $\times 22$
High Flexibility	5	1,698,693,120	163,074,539,520	$\approx \times 96$

Key Findings

1. Operational Feasibility: Empirical data confirm that the theoretical chromatic minimum of three rooms $(χ = 3)$ is both necessary and sufficient for practical scheduling, regardless of conference size $(n \geq 2)$ .
2. Exponential Flexibility Growth: Using only the minimum rooms, scheduling flexibility grows exponentially. Each additional team doubles the number of valid schedules, as dictated by the exact recurrence $P_{n} (3) = 2 . P_{n - 1} (3)$ , which is demonstrated by the data in Table 3.
3. Exponential Resource Leverage: Marginal increases in resources yield disproportionately large gains in flexibility. Crucially, the gain factor achieved by adding one room is exactly the value of the chromatic transition polynomial $ψ (k)$ , directly translating a graph-theoretic invariant into a concrete measure of operational efficiency, as quantified in Table 4.

4.5.6 Model Interpretation for Decision Support

This framework translates the mathematical constructs of $F_{n} \times P_{2}$ and its chromatic polynomial into actionable insights for conference planners. Table 5 provides the complete, systematic mapping that underpins this translation, linking each graph-theoretic element to its concrete scheduling counterpart and its direct practical significance for decision-makers.

Table 5. From graph elements to scheduling decisions.

Mathematical element	Scheduling representation	Practical significance for planners
Graph $F_{n} \times P_{2}$	Two-period conference model	Foundational structural framework
Central vertices	Coordinator sessions	Main scheduling constraint
Peripheral vertices	Team sessions	Team-specific resource requirement and task allocations
Horizontal edges	Concurrent session conflicts	Prevent concurrent room use for related sessions
Vertical edges	Cross-period session constraints	Assure no room is reused by the same team for sequential time slots
$P_{n} (k)$	Number of conflict-free schedules	Primary Flexibility Metric: Higher values indicate greater resilience to changes
$χ = 3$	Absolute minimum room requirement	Feasibility Guarantee: Establishes a strict lower bound for resource allocation
$ψ (k)$	Flexibility multiplier per team	Scalability Predictor: Enables forecasting of resource needs as the conference grows
Real root of $ψ (k)$ in $[2, 3]$	Theoretical $k$ threshold	Planning Threshold: Highlights the critical transition from infeasible ( $k = 2)$ to feasible $(k = 3)$ scheduling

The mapping elucidates the following core principles:

• The graph $F_{n} \times P_{2}$ serves as the foundational structural model for the entire two-period conference.
• Its vertices directly represent physical scheduling entities: central vertices correspond to coordinator sessions, while peripheral vertices represent team-specific sessions, thereby defining the core resource allocation requirements.
• The edges explicitly model the two critical types of scheduling conflicts: horizontal edges prevent concurrent room usage for related sessions, and vertical edges enforce the constraint that a team cannot reuse the same room across consecutive time periods.
• Most importantly, the key algebraic results—the chromatic polynomial $P_{n} (k)$ , the chromatic number $χ = 3$ , the transition polynomial $ψ (k)$ , and its real root in $[2, 3]$ —are transformed into practical planning tools. These tools quantitatively measure flexibility, guarantee minimum resource feasibility, predict scalability, and identify critical resource thresholds.

This structured translation equips planners with a clear, actionable methodology. It enables them to leverage the rigorous guarantees and predictive power of graph theory to make informed, concrete scheduling decisions and formulate robust, mathematically-grounded resource allocation strategies.

4.5.7 Model Limitations and Assumptions

1. All teams have identical scheduling constraints.
2. Meeting rooms are homogeneous and interchangeable.
3. No additional temporal constraints beyond the two-period framework.
4. The model assumes complete conflict graphs without probabilistic elements.

5. Discussion

5.1 Theoretical contribution

This work provides a complete algebraic characterization of the chromatic polynomial for $F_{n} \times P_{2}$ . The recurrence relation $P_{n} (k) = ψ (k) P_{n - 1} (k)$ establishes $ψ (k)$ as a new graph invariant that encodes the recursive constraint structure of this graph family. Crucially, the existence and constancy of this invariant stem from the structural symmetry and conditional independence of the recursively attached blocks $B_{n}$ —a property formalized in Lemma 4.1.4. Unlike well-documented results for standard Cartesian products,^9,12 this addresses a previously unexplored structure. Furthermore, the stability of the chromatic number at $χ = 3$ despite linear growth in order $(| V | = 4 n + 2$ ), highlights how restrictive local features determine global properties—a fundamental decoupling of structural scale from chromatic complexity.^6,7

5.2 Practical utility

The chromatic polynomial serves as a direct quantitative tool in our two-period conference scheduling model, building upon established graph-coloring paradigms.^1,13 The closed-form expression $P_{n} (k) = {[ψ (k)]}^{n - 2} P_{2} (k)$ reveals $ψ (k)$ as a flexibility multiplier: each additional room increases feasible schedules by a factor of $ψ (k)$ per team. For instance, moving from $k = 3$ to $k = 4$ rooms yields a $ψ (4) = 22$ -fold increase in valid allocations, as quantified in Table 4, transforming an abstract invariant into a practical decision-support metric.

From a computational perspective, the recurrence enables substantial efficiency gains. Computing $P_{n} (k)$ via exponentiation by squaring reduces the time complexity from exponential (under deletion–contraction) to $O (log n)$ . This demonstrates how structural insights into graph families can yield algorithmic improvements, rendering an otherwise intractable problem practical for large $n$ . In scheduling terms, evaluating $P_{100} (k)$ requires only about seven multiplications—feasible for real-time planning—whereas generic methods would be prohibitive.

Thus, the work bridges pure graph theory with applied computation: the same algebraic invariant that elucidates the chromatic structure also enables efficient enumeration, directly informing both algorithmic design and resource-allocation decisions in constrained environments.

5.3 Limitations and future directions

The model assumes identical resources and a two-period framework, suggesting natural extensions:

1. Generalization to $F_{n} \times P_{m}$ for $m > 2$ for multi-period scheduling.
2. Enhanced models incorporating heterogeneous resources via list-chromatic polynomials or weighted coloring models.
3. Theoretical connections between the asymptotic distribution of chromatic roots and phase transitions in 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, DeepSeek (deepseek.com) was used as a supplementary tool for language editing and algebraic verification. The authors critically reviewed all output and take full responsibility for the final work.

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

1. Kannan M, Sathiragavan M, Nivetha P, et al.: Graph coloring techniques in scheduling and resource allocation. Journal of Nonlinear Analysis and Optimization. 2024; 15(2-3).
2. Sazdanovic R, Scofield D: Structure of the chromatic polynomial. arXiv preprint arXiv:2411.15088. 2024.
3. Abbas Q, Mustafa G: Chromatic polynomial of a picture fuzzy graph with application in traffic light control. J. Appl. Math. Comput. 2024; 70(2): 1395–1418.
4. Arif NE, Hasni R, Alikhani S: Chromatic polynomials of certain polyphenylene dendrimers. J. Comput. Theor. Nanosci. 2012; 9(4): 560–563.
5. Dong F, Koh KM: Foundations of the chromatic polynomial. Handbook of the Tutte Polynomial and Related Topics. Chapman and Hall/CRC; 2022; 213–251.
6. Imrich W, Klavžar S, Rall DF: Topics in graph theory: Graphs and their Cartesian product. CRC Press; 2008.
7. Rall DF: Graph products and coloring.Nadjafi-Arani MJ, Khodkar A, editors. Topics in graph theory. Nova Science Publishers; 2012; 45–62.
8. Ismael WS, Copel HB, Kamdon SU: Chromatic polynomials of n-centipede and triangular snake TS_n graphs. Advances and Applications in Discrete Mathematics. 2023; 36: 1–9.
9. Christina M, Nigro M, Sasaki D: The chromatic polynomial of layered graphs. Mat. Contemp. 2025; 1: 1–8.
10. Yadav R, Sehgal A, Sehgal S, et al.: The chromatic polynomial of grid graph P₃ □ P_n. J. Appl. Math. Comput. 2024; 70(1): 619–637.
11. Nizami AR, Munir M, Khan AS, et al.: On chromaticity of ladder-type graphs. Science International (Lahore). 2016; 28(2): 829–836.
12. Pfaff TJ, Walker J: The chromatic polynomial of P₂ × P_n and C₃ × P_n. Missouri Journal of Mathematical Sciences. 2008; 20(3): 169–177.
13. Vyas MN, Hemalatha GB: Exam scheduling using graph coloring. Journal of Information Systems Engineering and Management. 2025; 10(24s).
14. Ali A, Chartrand G, Zhang P: Irregularity in graphs. Springer International Publishing; 2021.
15. Shi Y, Dehmer M, Li X, et al.: Graph polynomials. CRC Press; 2017.
16. Takahashi R: Expansions of the Potts model partition function along deletions and contractions. arXiv preprint arXiv:2405.07612. 2024.

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Version 2

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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.

Grant information

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

Article Versions (2)

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Published: 12 Jun 2026, 15:351

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© 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.

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M. Talab S and E. Arif N. Chromatic Polynomials of Fn×P2 Graphs: Algebraic Analysis and Scheduling Applications [version 1; peer review: 2 approved with reservations]. F1000Research 2026, 15:351 (https://doi.org/10.12688/f1000research.176896.1)

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Open Peer Review

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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.

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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
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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
Report a concern

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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.

Back to all reports

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.

View more View less

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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[2] 2. Sazdanovic R, Scofield D: Structure of the chromatic polynomial. arXiv preprint arXiv:2411.15088. 2024.

[3] 3. Abbas Q, Mustafa G: Chromatic polynomial of a picture fuzzy graph with application in traffic light control. J. Appl. Math. Comput. 2024; 70(2): 1395–1418.

[4] 4. Arif NE, Hasni R, Alikhani S: Chromatic polynomials of certain polyphenylene dendrimers. J. Comput. Theor. Nanosci. 2012; 9(4): 560–563.

[5] 5. Dong F, Koh KM: Foundations of the chromatic polynomial. Handbook of the Tutte Polynomial and Related Topics. Chapman and Hall/CRC; 2022; 213–251.

[6] 6. Imrich W, Klavžar S, Rall DF: Topics in graph theory: Graphs and their Cartesian product. CRC Press; 2008.

[7] 7. Rall DF: Graph products and coloring.Nadjafi-Arani MJ, Khodkar A, editors. Topics in graph theory. Nova Science Publishers; 2012; 45–62.

[8] 8. Ismael WS, Copel HB, Kamdon SU: Chromatic polynomials of n-centipede and triangular snake TS_n graphs. Advances and Applications in Discrete Mathematics. 2023; 36: 1–9.

[9] 9. Christina M, Nigro M, Sasaki D: The chromatic polynomial of layered graphs. Mat. Contemp. 2025; 1: 1–8.

[10] 10. Yadav R, Sehgal A, Sehgal S, et al.: The chromatic polynomial of grid graph P₃ □ P_n. J. Appl. Math. Comput. 2024; 70(1): 619–637.

[11] 11. Nizami AR, Munir M, Khan AS, et al.: On chromaticity of ladder-type graphs. Science International (Lahore). 2016; 28(2): 829–836.

[12] 12. Pfaff TJ, Walker J: The chromatic polynomial of P₂ × P_n and C₃ × P_n. Missouri Journal of Mathematical Sciences. 2008; 20(3): 169–177.

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[14] 14. Ali A, Chartrand G, Zhang P: Irregularity in graphs. Springer International Publishing; 2021.

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

Abstract

Background

Methods

Results

Conclusions

Keywords

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

3.2 Combinatorial derivation of the transition polynomial ψ(k)

Remark 3.2.1

3.3 Derivation of the recurrence and closed-form expression

3.4 Additional analytical and numerical validation

4. Results

4.1 Structural analysis of Fn×P2

Theorem 4.1.1:

Proof:

Theorem 4.1.2:

Proof:

Proposition 4.1.3:

Proof:

Lemma 4.1.4:

Proof:

Table 1. 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:

Corollary 4.3.3

Proof:

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

Remark 4.3.4

4.4 Numerical validation

4.5 Application: A chromatic model for hierarchical conference scheduling

Table 3. Conference scheduling with minimum rooms.

Table 4. Impact of room availability on scheduling flexibility.

Table 5. From graph elements to scheduling decisions.

5. Discussion

5.1 Theoretical contribution

5.2 Practical utility

5.3 Limitations and future directions

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 $n = 2, 3, 4, 5 .$

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

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

Table 1. Edge classification in $F_{n} \times P_{2}$ .

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