On the List Color Function Threshold

TL;DR

This paper investigates the list color function threshold of graphs, disproving a conjecture by showing that for certain graphs, the threshold can grow faster than the list chromatic number, with a lower bound proportional to the square root of the graph size.

Contribution

The authors demonstrate that no universal constant bounds the list color function threshold in terms of the list chromatic number, providing a counterexample with unbounded growth.

Findings

The threshold difference for complete bipartite graphs grows at least as C√l.

The result disproves Thomassen's conjecture on a universal bound.

The paper establishes a lower bound on the threshold difference for specific graphs.

Abstract

The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$. The list color function of graph $G$, denoted $P_{\ell}(G,m)$, is a list analogue of the chromatic polynomial that has been studied since the early 1990s, primarily through comparisons with the corresponding chromatic polynomial. It is known that for any graph $G$ there is a $k \in \mathbb{N}$ such that $P_\ell(G,m) = P(G,m)$ whenever $m \geq k$. The list color function threshold of $G$, denoted $\tau(G)$, is the smallest $k \geq \chi(G)$ such that $P_{\ell}(G,m) = P(G,m)$ whenever $m \geq k$. In 2009, Thomassen asked whether there is a universal constant $\alpha$ such that for any graph $G$, $\tau(G) \leq \chi_{\ell}(G) + \alpha$, where $\chi_{\ell}(G)$ is the list chromatic number of $G$. We show that the answer to this question is no by proving that there exists a constant $C$ such that $\tau(K_{2,l}) - \chi_{\ell}(K_{2,l}) \ge C\sqrt{l}$ for $l \ge 16$.