The greedy coloring algorithm shows that a graph of maximum degree at most Δ has chromatic number at most $\mathrm{\Delta }+1$, and this is tight for cliques. Much attention has been devoted to improving this “greedy bound” for graphs without large cliques. Brooks famously proved that this bound can be improved by one if $\mathrm{\Delta }\ge 3$ and the graph contains no clique of size $\mathrm{\Delta }+1$. Reed's Conjecture states that the “greedy bound” can be improved by k if the graph contains no clique of size $\mathrm{\Delta }+1-2k$. Johansson proved that the “greedy bound” can be improved by a factor of $\mathrm{\Omega }(\ln {(\mathrm{\Delta })}^{-1})$ or $\mathrm{\Omega }\left(\frac{\ln (\ln (\mathrm{\Delta }))}{\ln (\mathrm{\Delta })}\right)$ for graphs with no triangles or no cliques of any fixed size, respectively.

Notably missing is a linear improvement on the “greedy bound” for graphs without large cliques. In this paper, we prove that for sufficiently large Δ, if G is a graph with maximum degree at most Δ and no clique of size ω, then$\chi (G)\le 72\mathrm{\Delta }\sqrt{\frac{\ln (\omega )}{\ln (\mathrm{\Delta })}}.$ This implies that for sufficiently large Δ, if ${\omega }^{{(72c)}^2}\le \mathrm{\Delta }$ then $\chi (G)\le \mathrm{\Delta }/c$.

This bound actually holds for the list-chromatic and even the correspondence chromatic number (also known as DP-chromatic number). In fact, we prove what we call a “local version” of it, a result implying the existence of a coloring when the number of available colors for each vertex depends on local parameters, like the degree and the clique number of its neighborhood. We prove that for sufficiently large Δ, if G is a graph of maximum degree at most Δ and minimum degree at least ${\ln }^2(\mathrm{\Delta })$ with list-assignment L, then G is L-colorable if for each $v\in V(G)$,$|L(v)|\ge 72\mathrm{deg}(v)\cdot \min \{\sqrt{\frac{\ln (\omega (v))}{\ln (\mathrm{deg}(v))}},\frac{\omega (v)\ln (\ln (\mathrm{deg}(v)))}{\ln (\mathrm{deg}(v))},\frac{{\log }_2(\chi (v)+1)}{\ln (\mathrm{deg}(v))}\},$ where $\chi (v)$ denotes the chromatic number of the neighborhood of v and $\omega (v)$ denotes the size of a largest clique containing v. This simultaneously implies the linear improvement over the “greedy bound” and the two aforementioned results of Johansson.

贪心着色算法表明，最大度数最多为Δ的图最多有色数$\mathrm{\Delta }+1$，这对于派系来说很紧张。对于没有大团的图，人们已经投入了很多注意力来改进这种“贪婪界限”。布鲁克斯著名地证明了这个界限可以提高一，如果$\mathrm{\Delta }\ge 3$并且该图不包含大小集团$\mathrm{\Delta }+1$. 里德猜想指出，如果图不包含大小集团，“贪婪界”可以提高k$\mathrm{\Delta }+1-2ķ$. Johansson 证明了“贪婪界限”可以提高一个因子$\mathrm{\Omega }(\ln {(\mathrm{\Delta })}^{-1})$或者$\mathrm{\Omega }\left(\frac{\ln (\ln (\mathrm{\Delta }))}{\ln (\mathrm{\Delta })}\right)$对于没有三角形或没有任何固定大小的团的图，分别。

值得注意的是，对于没有大团的图，“贪婪界”的线性改进是缺失的。在本文中，我们证明对于足够大的 Δ，如果G是最大度数至多 Δ 且没有大小为ω的团的图，则$\chi (G)\le 72\mathrm{\Delta }\sqrt{\frac{\ln (\omega )}{\ln (\mathrm{\Delta })}}.$这意味着对于足够大的 Δ，如果${\omega }^{{(72C)}^2}\le \mathrm{\Delta }$然后$\chi (G)\le \mathrm{\Delta }/C$.

这个界限实际上适用于列表色数，甚至对应色数（也称为 DP 色数）。事实上，我们证明了所谓的“局部版本”，当每个顶点的可用颜色数量取决于局部参数（如其邻域的度数和团数）时，该结果暗示存在着色。我们证明，对于足够大的 Δ，如果G是最大度数最多为 Δ 且最小度数至少为${\ln }^2(\mathrm{\Delta })$使用列表分配L，则G是L可着色的，如果对于每个$v\in 五(G)$,$|\mathrm{大号}(v)|\ge 72\mathrm{度}(v)\cdot \mathrm{分钟}\{\sqrt{\frac{\ln (\omega (v))}{\ln (\mathrm{度}(v))}},\frac{\omega (v)\ln (\ln (\mathrm{度}(v)))}{\ln (\mathrm{度}(v))},\frac{{\mathrm{日志}}_2(\chi (v)+1)}{\ln (\mathrm{度}(v))}\},$在哪里$\chi (v)$表示v的邻域的色数，并且$\omega (v)$表示包含v的最大团的大小。这同时意味着对“贪婪界限”和 Johansson 的上述两个结果的线性改进。