Biclique-colouring verification complexity and biclique-colouring power graphs

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 bicliquecolouring, 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 k n is max(2k + 2 − n, 2) if n ≥ k + 1 and exactly n otherwise. The bicliquechromatic number of a power of a cycle C n 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 investi-