On complete subgraphs of color-critical graphs

A graph G is called k-critical if x(G) = k and x(G e) -C x(G) for each edge e of G, where x denotes the chromatic number. T. Gallai conjectured that every k-critical graph of order n contains at most n complete (k l)-subgraphs. In 1987, Stiebitz proved Gallai’s conjecture in the case k = 4, and in 1992 Abbott and Zhou proved Gallai’s conjecture for all k > 5. In their paper, Abbott and Zhou asked the following question: is it true that the number of complete (k 1)-subgraphs of any k-critical graph G of order n > k is at most n k + 3 (k > 5)? In this paper, we give a positive answer to the question above for the cases k = 5,6.