Bounding the List Color Function Threshold from Above

TL;DR

This paper investigates the list color function threshold of bipartite graphs, providing new bounds for complete bipartite graphs and establishing exact thresholds for specific small cases.

Contribution

It improves the upper bounds on the list color function threshold for $K_{2,n}$ and determines the exact thresholds for some small bipartite graphs.

Findings

$ au(K_{2,n}) ext{ is bounded above by } rac{n+2.05}{1.24}$

$ au(K_{2,3}) = ext{chromatic number}$

$ au(K_{2,4}) = au(K_{2,5}) = 3$

Abstract

The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$ for each $m \in \mathbb{N}$. In 1990, Kostochka and Sidorenko introduced the list color function of graph $G$, denoted $P_{\ell}(G,m)$, which is a list analogue of the chromatic polynomial. 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 $m \geq k$. It is known that for every graph $G$, $\tau(G)$ is finite, and in fact, $\tau(G) \leq (|E(G)|-1)/\ln(1+ \sqrt{2}) + 1$. It is also known that when $G$ is a cycle or chordal graph, $G$ is enumeratively chromatic-choosable which means $\tau(G) = \chi(G)$. A recent paper of Kaul et al. suggests that understanding the list color function threshold of complete bipartite graphs is essential to the study of the extremal behavior of $\tau$. In this paper we show that for any $n \geq 2$, $\tau(K_{2,n}) \leq \lceil (n+2.05)/1.24 \rceil$ which gives an improvement on the general upper bound for $\tau(G)$ when $G = K_{2,n}$. We also develop additional tools that allow us to show that $\tau(K_{2,3}) = \chi(K_{2,3})$ and $\tau(K_{2,4}) = \tau(K_{2,5}) = 3$.