Adaptable chromatic number of graph products

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 G k of G remains the same as that of G , the adaptable chromatic number of G k may increase with k . We conjecture that for all sufficiently large k the adaptable chromatic number of G k 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.