On-Line Coloring and Recursive Graph Theory

An on-line vertex coloring algorithm receives the vertices ofa graph in some externally determined order, and, whenever a new vertex is presented, the algorithm also learns to which of the previously presented vertices the new vertex is adjacent. As each vertex is received, the algorithm must make an irrevocable choice of a color to assign the new vertex, and it makes this choice without knowledge of future vertices. A class of graphs $r$ is said to be on-line $\chi$-bounded if there exists an on-line algorithm $A$ and a function $f$ such that $A$ uses at most $f(\omega(G))$ colors to properly color any graph $G$ in \Gamma. If $H$ is a graph, let Forb($H$) denote the class of graphs that do not induce $H$. The goal of this paper is to establish that Forb($T$) is on-line $\chi$-bounded for every radius-2 tree $T$. As a corollary, the authors answer a question of Schmerl's; the authors show that every recursive cocomparability graph can be recursively colored with a number of colors that depends only on its clique number.