On-line coloring of perfect graphs
Lovász, Saks, and Trotter showed that there exists an on-line algorithm which will color any on-linek-colorable graph onn vertices withO(nlog(2k−3)n/log(2k−4)n) colors. Vishwanathan showed that at least Ω(logk−1n/k k ) 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-linek-colorable perfect graph onn vertices withn10k/loglogn 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.
Mathematics Subject Classification (1991)05 C
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- D. Bean: Effective coloration,J. Symbolic Logic 41 (1976), 469–480.Google Scholar
- A. Gyárfás, andJ. Lehel: On-line and first-fit coloring of graphs,J. of Graph Theory 12 (1988), 217–227.Google Scholar
- A. Gyárfás, andJ. Lehel: First-Fit and on-line chromatic number of families of graphs,Ars Combinatorica 29C (1990), 168–176.Google Scholar
- S. Irani: Coloring inductive graphs on-line,Proceedings of the 3lst Annual Symposium on the Foundations of Computer Science, (1990), 470–479.Google Scholar
- H. A. Kierstead: The linearity of First-Fit for coloring interval graphs,SIAM J. on Discrete Math.,1 (1988), 526–530.Google Scholar
- H. A. Kierstead, S. G. Penrice, andW. T. Trotter: First-Fit and on-line coloring of graphs which do not induceP 5,SIAM J. on Discrete Mathematics,8 (1995), 485–498.Google Scholar
- H. A. Kierstead, S. G. Penrice, andW.T. Trotter: On-line graph coloring and recursive graph theory,SIAM J. on Discrete Math. 7 (1994), 72–89.Google Scholar
- H. A. Kierstead, andW. T. Trotter: An extremal problem in recursive combinatorics,Congressus Numerantium,33 (1981), 143–153.Google Scholar
- L. Lovász, M. Saks, andW. T. Trotter: An online graph coloring algorithm with sublinear performance ratio,Discrete Math., (1989), 319–325.Google Scholar
- M. Szegedy: Private communication.Google Scholar
- S. Vishwanathan: Randomized online graph coloring,J. Algorithms,13 (1992), 657–669.Google Scholar