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On-line graph coloring of \({\mathbb{P}_5}\)-free graphs

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Abstract

Kierstead et al. (SIAM J Discret Math 8:485–498, 1995) have shown 1 that the competitive function of on-line coloring for \({\mathbb{P}}_5\) -free graphs (i.e., graphs without induced path on 5 vertices) is bounded from above by the exponential function \({\left( 4^{\chi (\mathbb{G})} - 1\right) / 3}\) . No nontrivial lower bound was known. In this paper we show the quadratic lower bound \(\tiny{\left( {\begin{array}{*{20}c} {{\chi ({\mathbb{G}}) + 1}} \\ {2} \\ \end{array} } \right) }\) . More precisely, we prove that \(\tiny{\left( {\begin{array}{*{20}c} {{\chi ({\mathbb{G}}) + 1}} \\ {2} \\ \end{array} } \right) }\) is the exact competitive function for (\({\mathbb{C}}_4, {\mathbb{P}}_5\))-free graphs. In this paper we also prove that 2\(\kappa({\mathbb{G})}\) - 1 is the competitive function of the best clique covering on-line algorithm for (\({\mathbb{C}}_4, {\mathbb{P}}_5\))-free graphs.

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  1. Theoretical Computer Science Department, Jagiellonian University, Gronostajowa 3, 30-387, Kraków, Poland

    Iwona Cieślik

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  1. Iwona Cieślik
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Cieślik, I. On-line graph coloring of \({\mathbb{P}_5}\)-free graphs. Acta Informatica 45, 79–91 (2008). https://doi.org/10.1007/s00236-007-0064-x

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  • DOI: https://doi.org/10.1007/s00236-007-0064-x

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