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arXiv · 2409.06063

Counting List Colorings of Unlabeled Graphs

Hemanshu Kaul·Jeffrey A. Mudrock
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Abstract

The classic enumerative functions for counting colorings of a graph $G$, such as the chromatic polynomial $P(G,k)$, do so under the assumption that the given graph is labeled. In 1985, Hanlon defined and studied the chromatic polynomial for an unlabeled graph $\mathcal{G}$, $P(\mathcal{G}, k)$. Determining $P(\mathcal{G}, k)$ amounts to counting colorings under the action of automorphisms of $\mathcal{G}$. In this paper, we consider the problem of counting list colorings of unlabeled graphs. We extend Hanlon's definition to the list context and define the unlabeled list color function, $P_\ell(\mathcal{G}, k)$, of an unlabeled graph $\mathcal{G}$. In this context, we pursue a fundamental question whose analogues have driven much of the research on counting list colorings and its generalizations: For a given unlabeled graph $\mathcal{G}$, does $P_\ell(\mathcal{G}, k) = P(\mathcal{G}, k)$ when $k$ is large enough? We show the answer to this question is yes for almost all graphs, in particular, for a large class of unlabeled graphs that includes point-determining graphs (also known as twin-free graphs, irreducible graphs, and mating graphs).

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BibTeXRIS

Hemanshu Kaul, Jeffrey A. Mudrock. 2026-09-17. Counting List Colorings of Unlabeled Graphs. https://arxiv.org/abs/2409.06063

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