This is a theoretical study in algebraic and combinatorial graph theory, analyzing the chromatic polynomial of the graph family G n = F n × 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 ≥ 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 × 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 × 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. ∎

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 ≠ 1 , y ≠ 1 , x ≠ y , z ≠ 2 , t ≠ 2 , z ≠ t ,

and x ≠ z , y ≠ 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 ≠ 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 ∉ { 1 , 2 } , then z ≠ 2 , while t ≠ 2 , r . Before imposing z ≠ 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 ≠ 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 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 . 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.

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.

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 ≥ 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 ≥ 2 . This proves both the first-order recurrence and the closed-form expression for F n × P 2 .

Theorem 4.1.1:

(Structural Properties of F n × P 2 )

For n ≥ 2 , the graph F n × P 2 possesses the following structural properties, which directly affect its chromatic behavior: 1.

| V | = 4 n + 2 .

Where the notation d ( η ) denotes ( η ) vertices of degree d .

Remark 4.1.2:

(Edge classification in F n × P 2 )

The edge count in Theorem 4.1.1 can be refined by classifying the edges of F n × 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,

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

which agrees with the edge count in Theorem 4.1.1.

Theorem 4.2.1:

(Recurrence Relation)

For all n ≥ 3 , the chromatic polynomial of F n × 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 .

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 ] .

Interpretation

The value | ψ ( k ) | is the exponential growth constant of the sequence { P n ( k ) } for fixed k with P 2 ( k ) ≠ 0 . In the combinatorial range k ∈ ℤ , k ≥ 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 ) is 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 .

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 ) ;

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.

The following sample calculations illustrate the agreement: •

Theorem 4.2.1: P 3 ( 3 ) = ψ ( 3 ) P 2 ( 3 ) = 2 · 24 = 48 .

Moreover, the recurrence gives the exact growth ratio for fixed k : P 5 ( 4 ) P 4 ( 4 ) = 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 ≥ 3 , that theorem gives lim n → ∞ P n ( k ) 1 n = ψ ( k ) .

For finite n , however, the quantity P n ( k ) 1 n need not be close to ψ ( k ) . For example, P 5 ( 4 ) 1 5 = ( 61,843,584 ) 1 5 ≈ 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 P n ( k ) P n − 1 ( k ) = ψ ( k ) , which is an exact consequence of the recurrence for n ≥ 3 . The asymptotic growth statement remains the limit result established in Theorem 4.3.2.

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 × 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 } ∪ { u i A , w i A , u i B , w i B : 1 ≤ i ≤ 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 × 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 , … , n and each period X ∈ { 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 × 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 ≥ 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 , 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 ) = 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.

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 ) .

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.

For instance, P 5 ( 4 ) P 5 ( 3 ) = 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,

P 5 ( 5 ) P 5 ( 3 ) = 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 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.

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;

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.

This work provides an explicit algebraic characterization of the chromatic polynomial of F n × 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 ≥ 3 , together with the closed-form expression P n ( k ) = [ ψ ( k ) ] n − 2 P 2 ( k ) , n ≥ 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 coloring ^{6 , 7} and on related structured families. ^{8 – 12} Compared with earlier Cartesian-product families such as P 2 × P n , C 3 × 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.

The structural results show that | V | = 4 n + 2 , | E | = 8 n + 1 , χ ( F n × 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 ≥ 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 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 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.

The present work focuses on F n × P 2 . A natural extension is F n × 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

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

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.

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