Gallai's problem on Dirac's construction

Abstract

It is thought that T. Gallai posed the following problem concerning a construction due to G.A. Dirac: suppose that a graph K consists of disjoint subgraphs G and H and a set of edges joining them. If each of G, H, and K are colour critical graphs, under what circumstances is it then true that the join of G and H is complete? (That is to say, when is every vertex of G joined to every vertex of H?). In many cases the answer is affirmative, e.g. when G and H are both complete, as noted by Gallai. We present a small selection of graphs where the join is not complete.