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).
From: Jeffrey Mudrock [view email]
[v1]
Mon, 9 Sep 2024 20:53:21 UTC (13 KB)
[v2]
Tue, 3 Feb 2026 01:04:49 UTC (13 KB)
[v3]
Thu, 17 Sep 2026 15:27:19 UTC (13 KB)
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