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 A using (O) over tilde(Delta(1-2/k)) colors. This algorithm leads to an algorithm for k-colorable graph using (O) over tilde (n(1-3/(k+l))) colors. This improved Wigderson's algorithm, which uses O(n(1-1/(k-1))) colors, containing as a subroutine an algorithm using (Delta + 1) colors for graphs with maximum degree A. It is easy to imagine that an algorithm which uses less colors in terms of A 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 (O) over tilde(Delta(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.