A colouring of the vertices of a graph (or hypergraph) G is adapted to a given colouring of the edges of G if no edge has the same colour as both (or all) its vertices. The adaptable chromatic number of G is the smallest integer k such that each edge-colouring of G by colours 1,2,…,k admits an adapted vertex-colouring of G by the same colours 1,2,…,k. (The adaptable chromatic number is just one more than a previously investigated notion of chromatic capacity.) The adaptable chromatic number of a graph G is smaller than or equal to the ordinary chromatic number of G. While the ordinary chromatic number of all (categorical) powers Gk of G remains the same as that of G, the adaptable chromatic number of Gk may increase with k. We conjecture that for all sufficiently large k the adaptable chromatic number of Gk equals the chromatic number of G. When G is complete, we prove this conjecture with k≥4, and offer additional evidence suggesting it may hold with k≥2. We also discuss other products and propose several open problems.
