Counting subgraphs of coloring graphs using shadow graphs

TL;DR

This work introduces shadow graphs as a constructive tool to compute the subgraph counting polynomials $\pi_G^{(H)}(k)$, which count induced copies of $H$ inside the $k$-coloring graph $\mathcal{C}_k(G)$. By systematically replacing selected vertices with shadow structures and aggregating over state maps, the authors express $\pi_G^{(H)}(k)$ as combinations of chromatic polynomials of shadow graphs, enabling explicit formulas for diverse $H$, including $Q_d$ on trees. They derive closed forms for hypercube polynomials on trees, reveal connections to the generalized degree sequence and the chromatic symmetric function, and show that the generalized degree sequence is not a complete invariant. Finally, they provide a counterexample to a conjecture by AKLR25 showing that equal chromatic-pair polynomials do not guarantee equal chromatic polynomials, illustrating the method’s computational power and suggesting further links to graph invariants like the CSF.

Abstract

Given a graph $G$, the $k$-coloring graph $\mathcal{C}_k(G)$ is constructed by selecting proper $k$-colorings of $G$ as vertices, with an edge between two colorings if they differ in the color of exactly one vertex. The number of vertices in $\mathcal{C}_k(G)$ is the famous chromatic polynomial of $G$. Asgarli, Krehbiel, Levinson and Russell showed that for any subgraph $H$, the number of induced copies of $H$ in $\mathcal{C}_k(G)$ is a polynomial function in $k$. Hogan, Scott, Tamitegama, and Tan found a shorter proof for polynomiality of these chromatic $H$-polynomials. In this paper, we provide a method of constructing these polynomials explicitly in terms of chromatic polynomials of shadow graphs. We illustrate the practicality of our formulas by computing an explicit formula for $H$-polynomial for trees when $H=Q_d$ is an arbitrary hypercube, a task which does not seem approachable from previous methods. The coefficients of the resulting polynomials feature generalized degree sequences introduced by Crew. In the special case when $H=P_2$, the corresponding polynomial is dubbed the chromatic pairs polynomial. We present a pair of graphs $G_1$ and $G_2$ sharing the same chromatic pairs polynomial but different chromatic polynomials, disproving a conjecture raised by Asgarli, Krehbiel, Levinson and Russell.

Figures (13)

• Figure 1: Graphs $G_1$ and $G_2$ used as a counterexample to Conjecture 5.2.
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Theorems & Definitions (18)

• Theorem 1.1
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• proof
• proof