Counting List Colorings of Unlabeled Graphs

TL;DR

The paper studies counting list colorings of unlabeled graphs by defining the unlabeled list color function $P_ll(G,k)$ and examining when it agrees with the unlabeled chromatic polynomial $P(G,k)$ for large $k$. It extends Hanlon's unlabeled framework to the list-coloring setting, deriving lower bounds for the number of equivalence classes of colorings under automorphisms and analyzing the impact of graph structure on these counts. The main result shows that all unlabeled connected point-determining graphs satisfy $P_ll(G,k)=P(G,k)$ for sufficiently large $k$, and this property is preserved under certain graph-operations such as adding two universal vertices. These findings advance understanding of when unlabeled list-coloring counts coincide with labeled/unlabeled chromatic polynomials, with implications for asymptotic enumeration and orbit-counting in unlabeled graph coloring.

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).

Theorems & Definitions (22)

• Theorem 1: DZ22
• Theorem 2: H85
• Lemma 3: H85
• Lemma 5
• Proposition 6
• Conjecture 7
• Theorem 8
• Corollary 9
• Theorem 10
• Proposition 11