Lovász, Saks, and Trotter showed that there exists an on-line algorithm which will color any on-line k-colorable graph on n vertices with O(n log(2k-3) n/log(2k-4) n) colors. Vishwanathan showed that at least Ω(logk-1 n/kk) colors are needed. While these remain the best known bounds, they give a distressingly weak approximation of the number of colors required. In this article we study the case of perfect graphs. We prove that there exists an on-line algorithm which will color any on-line k-colorable perfect graph on n vertices with n10k/log log n colors and that Vishwanathan's techniques can be slightly modified to show that his lower bound also holds for perfect graphs. This suggests that Vishwanathan's lower bound is far from tight in the general case.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics