Adaptable chromatic number of graph products

A colouring of the vertices of a graph (or hypergraph) GG is adapted   to a given colouring of the edges of GG if no edge has the same colour as both (or all) its vertices. The adaptable chromatic number   of GG is the smallest integer kk such that each edge-colouring of GG by colours 1,2,…,k1,2,…,k admits an adapted vertex-colouring of GG by the same colours 1,2,…,k1,2,…,k. (The adaptable chromatic number is just one more than a previously investigated notion of chromatic capacity  .) The adaptable chromatic number of a graph GG is smaller than or equal to the ordinary chromatic number of GG. While the ordinary chromatic number of all (categorical) powers GkGk of GG remains the same as that of GG, the adaptable chromatic number of GkGk may increase with kk. We conjecture that for all sufficiently large kk the adaptable chromatic number of GkGk equals the chromatic number of GG. When GG is complete, we prove this conjecture with k≥4k≥4, and offer additional evidence suggesting it may hold with k≥2k≥2. We also discuss other products and propose several open problems.