First-Fit coloring of bounded tolerance graphs
Discrete Applied Mathematics
On-line coloring of h-free bipartite graphs
CIAC'06 Proceedings of the 6th Italian conference on Algorithms and Complexity
On the online track assignment problem
Discrete Applied Mathematics
Radius two trees specify χ-bounded classes
Journal of Graph Theory
On first-fit coloring of ladder-free posets
European Journal of Combinatorics
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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.