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arXiv · 1203.2543

Biclique-colouring verification complexity and biclique-colouring power graphs

Hélio B. Macêdo Filho·Simone Dantas·Raphael C. S. Machado·Celina M. H. de Figueiredo
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Abstract

Biclique-colouring is a colouring of the vertices of a graph in such a way that no maximal complete bipartite subgraph with at least one edge is monochromatic. We show that it is coNP-complete to check whether a given function that associates a colour to each vertex is a biclique-colouring, a result that justifies the search for structured classes where the biclique-colouring problem could be efficiently solved. We consider biclique-colouring restricted to powers of paths and powers of cycles. We determine the biclique-chromatic number of powers of paths and powers of cycles. The biclique-chromatic number of a power of a path P_{n}^{k} is max(2k + 2 - n, 2) if n >= k + 1 and exactly n otherwise. The biclique-chromatic number of a power of a cycle C_n^k is at most 3 if n >= 2k + 2 and exactly n otherwise; we additionally determine the powers of cycles that are 2-biclique-colourable. All proofs are algorithmic and provide polynomial-time biclique-colouring algorithms for graphs in the investigated classes.

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Hélio B. Macêdo Filho, Simone Dantas, Raphael C. S. Machado, Celina M. H. de Figueiredo. 2013-04-02. Biclique-colouring verification complexity and biclique-colouring power graphs. https://arxiv.org/abs/1203.2543

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