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On-line coloring of $I_s$-free graphs

Iwona Cieslik 1 Marcin Kozik 1 Piotr Micek 1
1 Algorithmics Research Group
UJ - Jagiellonian University [Krakow]
Abstract : An on-line vertex coloring algorithm receives vertices of a graph in some externally determined order. Each new vertex is presented together with a set of the edges connecting it to the previously presented vertices. As a vertex is presented, the algorithm assigns it a color which cannot be changed afterwards. The on-line coloring problem was addressed for many different classes of graphs defined in terms of forbidden structures. We analyze the class of $I_s$-free graphs, i.e., graphs in which the maximal size of an independent set is at most $s-1$. An old Szemerédi's result implies that for each on-line algorithm A there exists an on-line presentation of an $I_s$-free graph $G$ forcing A to use at least $\frac{s}{2}χ ^{(G)}$ colors. We prove that any greedy algorithm uses at most $\frac{s}{2}χ^{(G)}$ colors for any on-line presentation of any $I_s$-free graph $G$. Since the class of co-planar graphs is a subclass of $I_5$-free graphs all greedy algorithms use at most $\frac{5}{2}χ (G)$ colors for co-planar $G$'s. We prove that, even in a smaller class, this is an almost tight bound.
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Iwona Cieslik, Marcin Kozik, Piotr Micek. On-line coloring of $I_s$-free graphs. Computational Logic and Applications, CLA '05, 2005, Chambéry, France. pp.61-68. ⟨hal-01183336⟩

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