Better coloring of 3-colorable graphs
Ken-ichi Kawarabayashi, Mikkel Thorup, Hirotaka Yoneda

TL;DR
This paper introduces a new combinatorial algorithm and combines it with SDP techniques to improve the bounds on coloring 3-colorable graphs, reducing the number of colors needed to approximately n^{0.19747}.
Contribution
It presents a novel combinatorial approach for graphs with high minimum degree and integrates it with SDP methods to improve coloring bounds for 3-colorable graphs.
Findings
Achieves coloring with approximately n^{0.19747} colors.
Improves previous bounds from n^{0.19996} to n^{0.19747}.
Provides a polynomial-time algorithm for graphs with minimum degree above √n.
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 , we can, in polynomial time, make progress towards a -coloring for some . We balance our main result with the best-known semi-definite(SDP) approach which we use for degrees below . As a result, we show that colors suffice for coloring 3-colorable graphs. This improves on the previous best bound of by Kawarabayashi and Thorup in 2017.
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Taxonomy
TopicsGraph Labeling and Dimension Problems
