New Approximation Algorithms for Graph Coloring

Abstract

The problem of coloring a graph with the minimum number of colorsis well known to be NP-hard, even restricted tok-colorable graphs for constantk ≥ 3. This paper explores theapproximation problem of coloringk-colorable graphs with as fewadditional colors as possible in polynomial time, with special focus onthe case of k = 3. The previous best upper bound on the number of colors needed forcoloring 3-colorable n-vertex graphsin polynomial time was [Formula Omitted] colors by Berger and Rompel, improving a bound of [Formula Omitted] colors by Wigderson. This paper presents an algorithmto color any 3-colorable graph with [Formula Omitted] colors, thus breaking an“O1994)barrier”. The algorithm given here is based on examiningsecond-order neighborhoods of vertices, rather than just immediateneighborhoods of vertices as in previous approaches. We extend ourresults to improve the worst-case bounds for coloringk-colorable graphs for constantk > 3 as well. © 1994, ACM. All rights reserved.