Counting subgraphs of coloring graphs

TL;DR

This work generalizes the chromatic polynomial by counting induced subgraphs $H$ within the $k$-coloring graph $\\mathcal{C}_k(G)$ via the chromatic $H$-polynomial $\\pi_G^{(H)}(k)$. It proves polynomiality in $k$ for large enough $k$ and presents a concrete instance, the chromatic pairs polynomial $\\pi_G^{(P_2)}(k)$, counting edges in coloring graphs. The authors derive explicit formulas for several graph families (null, complete, trees, cycles, pseudotrees) and establish that $\\pi_G^{(P_2)}$ provides refined invariants for trees, uniquely determining degree sequences, with further results on coefficients and special cases. They also explore cliques, cycles, and higher-dimensional hypercube counts as additional invariants, discussing limitations and conjecturing that the family of all $H$-polynomials may determine the base graph. A recent advancement shows polynomiality for all $k$ (not just large $k$), strengthening the theoretical foundation and supporting the view that coloring graphs encode complete information about the original graph through these polynomials.

Abstract

The chromatic polynomial $π_{G}(k)$ of a graph $G$ can be viewed as counting the number of vertices in a family of coloring graphs $\mathcal C_k(G)$ associated with (proper) $k$-colorings of $G$ as a function of the number of colors $k$. These coloring graphs can be understood as a reconfiguration system. We generalize the chromatic polynomial to $π_G^{(H)}(k)$, counting occurrences of arbitrary induced subgraphs $H$ in these coloring graphs, and we prove that these functions are polynomial in $k$. In particular, we study the chromatic pairs polynomial $π_{G}^{(P_2)}(k)$, which counts the number of edges in coloring graphs, corresponding to the number of pairs of colorings that differ on a single vertex. We show two trees share a chromatic pairs polynomial if and only if they have the same degree sequence, and we conjecture that the chromatic pairs polynomial refines the chromatic polynomial in general. We also instantiate our polynomials with other choices of $H$ to generate new graph invariants.

Figures (14)

• Figure 1: $\mathcal{C}_3(P_3)$, the 3-coloring graph for $P_3$, with vertex labels indicating their underlying colorings
• Figure 2: The induced $P_3$ in $\mathcal{C}_3(P_3)$ generated by $(\{v_1,v_3\},C)$, for $C=\{c_1,c_2,c_3\}$ as in Eqs. \ref{['eq:c_example']}
• Figure 3: Explaining $\sigma$ and $\tau$ in terms of cycles (Eqs. \ref{['eq:edgecycle']} and \ref{['eq:tau']})
• Figure 4: Colorings of $P_n$ that violate $\{1,2\}$ restraints for both leaves
• Figure 5: Counting the $\binom{k}{2}\frac{\pi_{C_4}(k)}{k} (k-2)^2(k-1)^3$ occurrences of $P_2$ due to non-cycle vertex $v_8$

Theorems & Definitions (35)

• Theorem : Theorem \ref{['thm:general']}
• Theorem : Theorem \ref{['thm:lowcoeffs']}
• Conjecture : Conjecture \ref{['conj:edge-implies-chromatic']}
• Conjecture : Conjecture \ref{['conj:col-graphs-determine-G']}
• Theorem 3.1
• proof
• Remark 3.2
• Corollary 3.3
• Theorem 5.1
• proof