Let G be a graph with chromatic number χ, maximum degree Δ and clique number ω. Reed's conjecture states that $\chi \le \lceil (1-\epsilon )(\mathrm{\Delta }+1)+\epsilon \omega \rceil$ for all $\epsilon \le 1/2$. It was shown by King and Reed that, provided Δ is large enough, the conjecture holds for $\epsilon \le 1/130,000$. In this article, we show that the same statement holds for $\epsilon \le 1/26$, thus making a significant step towards Reed's conjecture. We derive this result from a general technique to bound the chromatic number of a graph where no vertex has many edges in its neighbourhood. Our improvements to this method also lead to improved bounds on the strong chromatic index of general graphs. We prove that ${\chi }_s^{\prime }(G)\le 1.835\mathrm{\Delta }{(G)}^2$ provided $\mathrm{\Delta }(G)$ is large enough.

设G为色数为χ，最大度数为 Δ ，团数为ω的图。里德猜想表明$\chi \le \lceil (1-\epsilon )(\mathrm{\Delta }+1)+\epsilon \omega \rceil$对所有人$\epsilon \le 1/2$. King 和 Reed 证明，只要 Δ 足够大，猜想对于$\epsilon \le 1/130,000$. 在本文中，我们证明了同样的陈述适用于$\epsilon \le 1/26$，从而朝着里德猜想迈出了重要的一步。我们从一种通用技术中得出这个结果，该技术可以限制图的色数，其中没有顶点在其邻域中有很多边。我们对这种方法的改进也导致了对一般图的强色指数的改进。我们证明${\chi }_s^{\text{'}}(G)\le 1.835\mathrm{\Delta }{(G)}^2$假如$\mathrm{\Delta }(G)$足够大。