We give a short proof of a bound on the list chromatic number of graphs $G$ of maximum degree $\Delta$ where each neighbourhood has density at most $d$, namely $\chi_\ell(G) \le (1+o(1)) \frac{\Delta}{\ln \frac{\Delta}{d+1}}$ as $\frac{\Delta}{d+1} \to \infty$. This bound is tight up to an asymptotic factor $2$, which is the best possible barring a breakthrough in Ramsey theory, and strengthens results due to Vu, and more recently Davies, P., Kang, and Sereni. Our proof relies on the first moment method, and adapts a clever counting argument developed by Rosenfeld in the context of non-repetitive colourings. As a final touch, we show that our method provides an asymptotically tight lower bound on the number of colourings of locally sparse graphs.
翻译:我们简短地证明在列表色谱数中,每个街区的密度最高为$G$=Delta$(G)$chi ⁇ ell(G)\le(1+o(1))\frac\Delta=Delta=d+1 ⁇ $(Frac\Delta=Delta ⁇ d+1}}\ to\ fty$(美元)的绑定数字。这个绑紧到一个无症状系数$$2,这是拉姆西理论中取得突破的最好办法,并且由于Vu和最近的Davies、P.、Kang和Sereni而加强了结果。我们的证据依靠的是第一个时刻的方法,并调整了罗森菲尔德在非恒定色颜色方面开发的聪明计数参数。最后,我们显示我们的方法对本地稀亮图的颜色数量提供了同样紧凑的绑定。