In 1998, Reed conjectured that every graph G satisfies $\chi (G)\le \lceil \frac{1}{2}(\mathrm{\Delta }(G)+1+\omega (G))\rceil$, where $\chi (G)$ is the chromatic number of G, $\mathrm{\Delta }(G)$ 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 $\epsilon >0$ such that $\chi (G)\le (1-\epsilon )(\mathrm{\Delta }(G)+1)+\epsilon \omega (G)$. 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, $|L(v)|\ge \lceil \frac{1}{2}(d(v)+1+\omega (v))\rceil$, 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 $\epsilon \le \frac{{\alpha }^2}{1350}$, 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 $(1+\epsilon )k$. This implies that for every $\alpha >0$, there exists $\epsilon >0$ such that if G is a graph with $\omega (G)\le (\frac{1}{2}-\alpha )\mathrm{mad}(G)$, where $\mathrm{mad}(G)$ is the maximum average degree of G, then ${\chi }_{\ell }(G)\le \lceil (1-\epsilon )(\mathrm{mad}(G)+1)+\epsilon \omega (G)\rceil$. It also yields an improvement on the best known upper bound for the chromatic number of $K_t$-minor free graphs for large t, by a factor of .99982.

在1998年，里德（Reed）猜想每张图G都满足$\chi （G）\le \lceil \frac{\mathrm{1个}}{2}（\mathrm{\Delta }（G）+\mathrm{1个}+\omega （G））\rceil$，在哪里 $\chi （G）$是G的色数，$\mathrm{\Delta }（G）$是G的最大程度，并且$\omega （G）$是G的集团编号。作为他猜想的证据，他证明了它的“ε形式”，即存在一些$\epsilon >0$ 这样 $\chi （G）\le （\mathrm{1个}-\epsilon ）（\mathrm{\Delta }（G）+\mathrm{1个}）+\epsilon \omega （G）$。很自然地会问里德的猜想或它的ε形式是否适用于列表色数。在本文中，我们考虑了里德猜想的列表颜色版本的“本地版本”。即，我们推测，如果ģ与列表分配的曲线图大号，使得对于每个顶点v的ģ，$|\mathrm{大号}（v）|\ge \lceil \frac{\mathrm{1个}}{2}（d（v）+\mathrm{1个}+\omega （v））\rceil$，在哪里 $d（v）$是v的次数，$\omega （v）$是包含v的最大派系的大小，则G是L可着色的。我们的主要结果是，在一些温和的假设下，该猜想的“ε形式”是正确的。

使用这个结果，我们还证明了k临界图的密度显着改善的下界，其集团数小于$ķ/2$， 如下。对于每个$\alpha >0$如果 $\epsilon \le \frac{{\alpha }^2}{1350}$，则如果G是某个k列表分配L的L临界图，则使得$\omega （G）<（\frac{\mathrm{1个}}{2}-\alpha ）ķ$并且k足够大，那么G至少具有平均度$（\mathrm{1个}+\epsilon ）ķ$。这意味着对于每个$\alpha >0$， 那里存在 $\epsilon >0$这样，如果G是具有$\omega （G）\le （\frac{\mathrm{1个}}{2}-\alpha ）\mathrm{狂}（G）$，在哪里 $\mathrm{狂}（G）$是G的最大平均度，则${\chi }_{\ell }（G）\le \lceil （\mathrm{1个}-\epsilon ）（\mathrm{狂}（G）+\mathrm{1个}）+\epsilon \omega （G）\rceil$。它还可以改善最著名的色度数上限。${ķ}_{Ť}$大t的次要免费图，因子为.99982。