Biclique-colouring verification complexity and biclique-colouring power graphs

TL;DR

This paper investigates the complexity of biclique-colouring verification, proves its coNP-completeness, and provides polynomial-time algorithms for specific classes like powers of paths and cycles, including exact biclique-chromatic numbers.

Contribution

It establishes the coNP-completeness of biclique-colouring verification and determines biclique-chromatic numbers for powers of paths and cycles, with efficient algorithms for these classes.

Findings

Verification of biclique-colouring is coNP-complete.

Exact biclique-chromatic numbers are determined for powers of paths and cycles.

Polynomial-time algorithms are provided for these classes.

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.