Counting List Colorings of Unlabeled Graphs

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 a large class of unlabeled graphs that include point-determining graphs (also known as twin-free graphs, irreducible graphs, and mating graphs).

💡 Research Summary

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The paper investigates the enumeration of list colorings for unlabeled graphs, extending the classical chromatic polynomial framework to the list‑coloring setting. For a labeled graph (G), the chromatic polynomial (P(G,k)) counts proper (k)-colorings, while the list‑color function (P_{\ell}(G,k)) (introduced by Kostochka and Sidorenko) counts the minimum number of proper colorings over all possible (k)-list assignments. Hanlon (1985) defined an “unlabeled chromatic polynomial” (P(\mathcal{G},k)) for an isomorphism class (\mathcal{G}) by counting colorings up to the action of the automorphism group (\operatorname{Aut}(\mathcal{G})).

The authors define the unlabeled list‑color function (P_{\ell}(\mathcal{G},k)) analogously: for a fixed representative (G) of the class (\mathcal{G}) and a (k)-list assignment (L), two proper (L)-colorings are considered equivalent if an automorphism of (G) maps one to the other. The function (P_{\ell}(\mathcal{G},k)) is the minimum number of equivalence classes over all possible (k)-assignments. By applying Burnside’s Lemma to the trivial list assignment (every vertex receives the full set (