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This paper presents a polynomial-time algorithm to color any 3-colorable n-node graph with O(n2/5 log8/5 n) colors, improving the best previously known bound of O(√n/√logn) colors. By reducing the number of colors needed to color a 3-colorable graph, the algorithm also improves the bound for k-coloring for fixed k ≥ 3 from the previous O((n/log n)1-1/(k-1)) colors to O(n1-1/(k-4/3) log8/5 n) colors. An extension of the algorithm further improves the bounds. Precise values appear in a table at the end of this paper.
Avrim Blum (Sun,) studied this question.
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