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Online Coloring of Bipartite Graphs with and without Advice

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Abstract

In the online version of the well-known graph coloring problem, the vertices appear one after the other together with the edges to the already known vertices and have to be irrevocably colored immediately after their appearance. We consider this problem on bipartite, i.e., two-colorable graphs. We prove that at least ⌊1.13746⋅log2(n)−0.49887⌋ colors are necessary for any deterministic online algorithm to be able to color any given bipartite graph on n vertices, thus improving on the previously known lower bound of ⌊log2 n⌋+1 for sufficiently large n.

Recently, the advice complexity was introduced as a method for a fine-grained analysis of the hardness of online problems. We apply this method to the online coloring problem and prove (almost) tight linear upper and lower bounds on the advice complexity of coloring a bipartite graph online optimally or using 3 colors. Moreover, we prove that \(O(\sqrt{n})\) advice bits are sufficient for coloring any bipartite graph on n vertices with at most ⌈log2 n⌉ colors.

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Acknowledgements

The authors would like to thank the anonymous referees for helpful suggestions. The research was partially funded by the SNF grant 200021–141089.

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Authors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Milano, Milan, Italy

    Maria Paola Bianchi

  2. Department of Computer Science, ETH Zurich, Zurich, Switzerland

    Hans-Joachim Böckenhauer, Juraj Hromkovič & Lucia Keller

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  1. Maria Paola Bianchi
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  2. Hans-Joachim Böckenhauer
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  3. Juraj Hromkovič
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  4. Lucia Keller
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Corresponding author

Correspondence to Hans-Joachim Böckenhauer.

Additional information

A preliminary version of this paper has been presented at the 18th Annual International Computing and Combinatorics Conference (COCOON 2012).

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Bianchi, M.P., Böckenhauer, HJ., Hromkovič, J. et al. Online Coloring of Bipartite Graphs with and without Advice. Algorithmica 70, 92–111 (2014). https://doi.org/10.1007/s00453-013-9819-7

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