The complexity of H-colouring of bounded degree graphs

Abstract

We investigate the complexity of the H-colouring problem restricted to graphs of bounded degree. The H-colouring problem is a generalization of the standard c-colouring problem, whose restriction to bounded degree graphs remains NP-complete, as long as c is smaller than the degree bound (otherwise we can use the theorem of Brooks to obtain a polynomial time algorithm). For H-colouring of bounded degree graphs, while it is also the case that most problems are NP-complete, we point out that, surprisingly, there exist polynomial algorithms for several of these restricted colouring problems. Our main objective is to propose a conjecture about the complexity of certain cases of the problem. The conjecture states that for graphs of chromatic number three, all situations which are not solvable by the colouring algorithm inherent in the theorem of Brooks are NP-complete. We motivate the conjecture by proving several supporting results.