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Colouring Clique-Hypergraphs of Circulant Graphs

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Abstract

A clique-colouring of a graph G is a colouring of the vertices of G so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, \({\mathcal{H}(G)}\) , of a graph G has V(G) as its set of vertices and the maximal cliques of G as its hyperedges. A vertex-colouring of \({\mathcal{H}(G)}\) is a clique-colouring of G. Determining the clique-chromatic number, the least number of colours for which a graph G admits a clique-colouring, is known to be NP-hard. In this work, we establish that the clique-chromatic number of powers of cycles is equal to two, except for odd cycles of size at least five, that need three colours. For odd-seq circulant graphs, we show that their clique-chromatic number is at most four, and determine the cases when it is equal to two. Similar bounds for the chromatic number of these graphs are also obtained.

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Authors and Affiliations

  1. Institute of Computing, University of Campinas, Campinas, Brazil

    C. N. Campos & C. P. de Mello

  2. Institute of Mathematics and Statistics, Fluminense Federal University, Niterói, Brazil

    S. Dantas

Authors
  1. C. N. Campos
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  2. S. Dantas
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  3. C. P. de Mello
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Correspondence to C. N. Campos.

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Partially supported by CNPq, FAPESP and FAPERJ.

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Campos, C.N., Dantas, S. & de Mello, C.P. Colouring Clique-Hypergraphs of Circulant Graphs. Graphs and Combinatorics 29, 1713–1720 (2013). https://doi.org/10.1007/s00373-012-1241-4

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  • DOI: https://doi.org/10.1007/s00373-012-1241-4

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