A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph $F$, there is a constant $c_F > 0$ such that if $G$ is an $F$-free graph of maximum degree $\Delta$, then $\chi(G) \leq c_F \Delta / \log\Delta$. Alon, Krivelevich, and Sudakov verified this conjecture for a class of graphs $F$ that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot, and Sereni that if $G$ is $K_{t,t}$-free, then $\chi(G) \leq (t + o(1)) \Delta / \log\Delta$ as $\Delta \to \infty$. We improve this bound to $(1+o(1)) \Delta/\log \Delta$, making the constant factor independent of $t$. We further extend our result to the DP-coloring setting (also known as correspondence coloring), introduced by Dvo\v{r}\'ak and Postle.

翻译：Alon、Krivelevich和Sudakov的注解指出,对于任何一张包括所有双面图在内的图表来说,美元总是以美元计价的,美元 > 0美元,因此,如果G$是最高度为$\Delta$的无美元图,那么,$\chi(G)\leq c_F\Delta/\log\Delta$。Alon、Krivelevich和Sudakov对包括所有双面图在内的某类图表的这一推论进行了核实。此外,Davies、Kang、Pirot和Sereni最近的工作显示,如果G$是$@t,t}$免费,然后$\chi(G)\leq (t+o(1))\delta/log\delta$,作为$\delta\to\infty$。我们把这个绑定的绑定值值值值值值值值,我们进一步扩展了Delta/Delta\logta$(我们所知道的彩色),通过建立固定的汇率。