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arXiv · 2406.00357

Better coloring of 3-colorable graphs

Ken-ichi Kawarabayashi·Mikkel Thorup·Hirotaka Yoneda
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Abstract

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.

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Ken-ichi Kawarabayashi, Mikkel Thorup, Hirotaka Yoneda. 2024-06-01. Better coloring of 3-colorable graphs. https://arxiv.org/abs/2406.00357

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