Counting List Colorings of Unlabeled Graphs

TL;DR

This paper extends the concept of counting graph colorings to unlabeled graphs by defining an unlabeled list color function and investigates when it equals the traditional chromatic polynomial for large k.

Contribution

It introduces the unlabeled list color function for graphs and proves its equality with the chromatic polynomial for a broad class of graphs when k is sufficiently large.

Findings

The unlabeled list color function is defined as an extension of Hanlon's chromatic polynomial.

For many graphs, including twin-free graphs, the list color function matches the chromatic polynomial for large k.

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