A local epsilon version of Reed's Conjecture

TL;DR

This paper explores a local, epsilon-based version of Reed's conjecture for list coloring, proving its validity under mild conditions and deriving new bounds on the density of certain critical graphs.

Contribution

It establishes an epsilon version of the local list-coloring Reed's conjecture and improves bounds on the density of k-critical graphs with small clique number.

Findings

Proves an epsilon version of the local Reed's conjecture for list coloring.

Provides lower bounds on the density of k-critical graphs with small clique number.

Connects the conjecture to bounds involving maximum average degree and clique number.

Abstract

In 1998, Reed conjectured that every graph $G$ satisfies $\chi(G) \leq \lceil \frac{1}{2}(\Delta(G) + 1 + \omega(G))\rceil$, where $\chi(G)$ is the chromatic number of $G$, $\Delta(G)$ is the maximum degree of $G$, and $\omega(G)$ is the clique number of $G$. As evidence for his conjecture, he proved an "epsilon version" of it, i.e. that there exists some $\varepsilon > 0$ such that $\chi(G) \leq (1 - \varepsilon)(\Delta(G) + 1) + \varepsilon\omega(G)$. It is natural to ask if Reed's conjecture or an epsilon version of it is true for the list-chromatic number. In this paper we consider a "local version" of the list-coloring version of Reed's conjecture. Namely, we conjecture that if $G$ is a graph with list-assignment $L$ such that for each vertex $v$ of $G$, $|L(v)| \geq \lceil \frac{1}{2}(d(v) + 1 + \omega(v))\rceil$, where $d(v)$ is the degree of $v$ and $\omega(v)$ is the size of the largest clique containing $v$, then $G$ is $L$-colorable. Our main result is that an "epsilon version" of this conjecture is true, under some mild assumptions. Using this result, we also prove a significantly improved lower bound on the density of $k$-critical graphs with clique number less than $k/2$, as follows. For every $\alpha > 0$, if $\varepsilon \leq \frac{\alpha^2}{1350}$, then if $G$ is an $L$-critical graph for some $k$-list-assignment $L$ such that $\omega(G) < (\frac{1}{2} - \alpha)k$ and $k$ is sufficiently large, then $G$ has average degree at least $(1 + \varepsilon)k$. This implies that for every $\alpha > 0$, there exists $\varepsilon > 0$ such that if $G$ is a graph with $\omega(G)\leq (\frac{1}{2} - \alpha)\mathrm{mad}(G)$, where $\mathrm{mad}(G)$ is the maximum average degree of $G$, then $\chi_\ell(G) \leq \left\lceil (1 - \varepsilon)(\mathrm{mad}(G) + 1) + \varepsilon \omega(G)\right\rceil$.