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Equitable colorings of Cartesian products of square of cycles and paths with complete bipartite graphs

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Abstract

A graph G is said to be equitably k-colorable if the vertex set of G can be divided into k independent sets for which any two sets differ in size at most one. The equitable chromatic number of G, \(\chi _{=}(G)\), is the minimum k for which G is equitably k-colorable. The equitable chromatic threshold of G, \(\chi _{=}^{*}(G)\), is the minimum k for which G is equitably \(k'\)-colorable for all \(k'\ge k\). In this paper, the exact values of \(\chi _{=}^{*}(G\Box H)\) and \(\chi _{=}(G\Box H)\) are obtained when G is the square of a cycle or a path and H is a complete bipartite graph.

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Acknowledgments

We thank the anonymous referee’s helpful suggestions for improving this paper substantial. This work is supported by NSFC for youth with code 61103073.

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Authors and Affiliations

  1. College of Mathematical Science, Tianjin Normal University, Tianjin, 300387, China

    Shasha Ma & Liancui Zuo

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  1. Shasha Ma
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  2. Liancui Zuo
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Correspondence to Liancui Zuo.

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Ma, S., Zuo, L. Equitable colorings of Cartesian products of square of cycles and paths with complete bipartite graphs. J Comb Optim 32, 725–740 (2016). https://doi.org/10.1007/s10878-015-9895-5

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