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arXiv · 2609.08540

When chromatic polynomials coincide with list-color functions: a threshold linear in the maximum degree

Meiqiao Zhang·Fengming Dong
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Abstract

Let $G$ be a simple graph with maximum degree $\Delta\ge 3$, and let $P(G,k)$ denote its chromatic polynomial. For each positive integer $k$, the list-color function $P_{\ell}(G,k)$ is the minimum number of $L$-colorings of $G$ over all $k$-assignments $L$. In this paper, we prove that $P_{\ell}(G,k)=P(G,k)$ for every integer $k\ge 23.41\Delta$. This gives a threshold for equality that is linear in the maximum degree and independent of the number of vertices or edges. It improves the known sufficient condition $k\ge |E(G)|-1$ for graphs with sufficiently many edges relative to their maximum degree.

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Meiqiao Zhang, Fengming Dong. 2026-09-08. When chromatic polynomials coincide with list-color functions: a threshold linear in the maximum degree. https://arxiv.org/abs/2609.08540

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