On-line coloringk-colorable graphs

We show that for anyk, there exists an on-line algorithm that will color anyk-colorable graph onn vertices withO(n ^1−1/k! ) colors. This improves the previous best upper bound ofO(nlog^(2k−3) n/log^(2k−4) n) due to Lovász, Saks, and Trotter. In the special casesk=3 andk=4 we obtain on-line algorithms that useO(n ^2/3log^1/3 n) andO(n ^5/6log^1/6 n) colors, respectively.