A local epsilon version of Reed's Conjecture
In 1998, Reed conjectured that every graph $G$ satisfies $χ(G) \leq \lceil \frac{1}{2}(Δ(G) + 1 + ω(G))\rceil$, where $χ(G)$ is the chromatic number of $G$, $Δ(G)$ is the maximum degree of $G$, and $ω(G)$ 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 $χ(G) \leq (1 - \varepsilon)(Δ(G) + 1) + \varepsilonω(G)$. 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 $G$, $|L(v)| \geq \lceil \frac{1}{2}(d(v) + 1 + ω(v))\rceil$, where $d(v)$ is the degree of $v$ and $ω(v)$ 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 $α> 0$, if $\varepsilon \leq \frac{α^2}{1350}$, then if $G$ is an $L$-critical graph for some $k$-list-assignment $L$ such that $ω(G) < (\frac{1}{2} - α)k$ and $k$ is sufficiently large, then $G$ has average degree at least $(1 + \varepsilon)k$. This implies that for every $α> 0$, there exists $\varepsilon > 0$ such that if $G$ is a graph with $ω(G)\leq (\frac{1}{2} - α)\mathrm{mad}(G)$, where $\mathrm{mad}(G)$ is the maximum average degree of $G$, then $χ_\ell(G) \leq \left\lceil (1 - \varepsilon)(\mathrm{mad}(G) + 1) + \varepsilon ω(G)\right\rceil$.