Counting subgraphs of coloring graphs using shadow graphs

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.