  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.
