We present a polynomial-time algorithm that colors any 3-colorable $n$-vertex graph using $O(n^{0.19539})$ colors, improving upon the previous best bound of $\widetilde{O}(n^{0.19747})$ by Kawarabayashi, Thorup, and Yoneda [STOC 2024]. Our result constitutes the first progress in nearly two decades on SDP-based approaches to this problem. The earlier SDP-based algorithms of Arora, Chlamtáč, and Charikar [STOC 2006] and Chlamtáč [FOCS 2007] rely on extracting a large independent set from a suitably "random-looking" second-level neighborhood, under the assumption that the KMS algorithm [Karger, Motwani, and Sudan, JACM 1998] fails to find one globally. We extend their analysis to third-level neighborhoods. We then come up with a new vector $5/2$-coloring, which allows us to extract a large independent set from some third-level neighborhood. The new vector coloring construction may be of independent interest.

翻译：我们提出了一种多项式时间算法，该算法能够使用$O(n^{0.19539})$种颜色对任意具有$n$个顶点的三可着色图进行染色，改进了Kawarabayashi、Thorup和Yoneda [STOC 2024]先前提出的$\widetilde{O}(n^{0.19747})$的最佳界限。我们的结果是近二十年来该问题基于SDP方法的首个进展。Arora、Chlamtáč和Charikar [STOC 2006]以及Chlamtáč [FOCS 2007]的早期基于SDP的算法依赖于从一个具有适当"随机性外观"的第二层邻域中提取一个大的独立集，其前提是假设KMS算法 [Karger, Motwani和Sudan, JACM 1998]无法在全局范围内找到一个这样的独立集。我们将他们的分析扩展到了第三层邻域。随后，我们提出了一种新的向量$5/2$染色方案，这使得我们能够从某个第三层邻域中提取一个大的独立集。这一新的向量染色构造方案可能具有独立的研究价值。