We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. This is one of the most challenging problems in graph algorithms. In this paper using Blum's notion of ``progress'', we develop a new combinatorial algorithm for the following: Given any 3-colorable graph with minimum degree $\ds>\sqrt n$, we can, in polynomial time, make progress towards a $k$-coloring for some $k=\sqrt{n/\ds}\cdot n^{o(1)}$. We balance our main result with the best-known semi-definite(SDP) approach which we use for degrees below $n^{0.605073}$. As a result, we show that $\tO(n^{0.19747})$ colors suffice for coloring 3-colorable graphs. This improves on the previous best bound of $\tO(n^{0.19996})$ by Kawarabayashi and Thorup in 2017.

翻译：我们研究在多项式时间内用尽可能少的颜色对三可着色图进行着色的问题。这是图算法中最具挑战性的问题之一。本文利用Blum提出的"进展"概念，针对以下问题提出了一种新的组合算法：对于任意最小度$\ds>\sqrt n$的三可着色图，我们可以在多项式时间内朝着$k$着色取得进展，其中$k=\sqrt{n/\ds}\cdot n^{o(1)}$。我们将主要结果与最著名的半定规划（SDP）方法相结合，后者适用于度数低于$n^{0.605073}$的情况。最终，我们证明$\tO(n^{0.19747})$种颜色足以对三可着色图进行着色。这改进了Kawarabayashi和Thorup在2017年提出的先前最佳界限$\tO(n^{0.19996})$。