On Approximation Algorithms for Coloring k-Colorable Graphs

Karger, Motwani and Sudan presented a graph coloring algorithm based on semidefinite programming, which colors any k-colorable graph with maximum degree ∆ using Õ(∆1−2/k) colors. This algorithm leads to an algorithm for k-colorable graph using Õ(n1−3/(k+1)) colors. This improved Wigderson’s algorithm, which uses O(n1−1/(k−1)) colors, containing as a subroutine an algorithm using (∆ + 1) colors for graphs with maximum degree ∆. It is easy to imagine that an algorithm which uses less colors in terms of ∆ leads to an algorithm which uses less colors in terms of n. In this paper, we consider this influence assuming that we have an algorithm which uses Õ(∆1−x/k) colors for 2 < x < 3. Specifically, we will show that the algorithms of Karger et al., of Blum and Karger and of Halperin et al. can be improved under this assumption. key words: graph coloring, approximation algorithms, NP-hard, maximum degree