A Note on Δ-Critical Graphs

A k-critical graph is a k-chromatic graph whose proper subgraphs are all (k - 1)-colourable. An old open problem due to Borodin and Kostochka asserts that for k >= 9, no k-critical graph G with k = Delta(G) exists, where Delta(G) denotes the maximum degree of G. We show that if a certain special list-colouring property holds for every 8-critical graph with Delta = 8 (which is true for the apparently only known example), then the Borodin-Kostochka Conjecture holds. We also briefly survey constructions of Delta-critical graphs with Delta <= 8, highlighting the apparent scarcity of such graphs once Delta exceeds 6.