On complete subgraphs of color-critical graphs Original Research Article

A graph G is called k-critical if χ(G) = k and χ(G − e) < χ(G) for each edge e of G, where χ denotes the chromatic number. T. Gallai conjectured that every k-critical graph of order n contains at most n complete (k − 1)-subgraphs. In 1987, Stiebitz proved Gallaiʹs conjecture in the case k = 4, and in 1992 Abbott and Zhou proved Gallaiʹs 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.