Complete and Almost Complete Minors in Double-Critical 8-Chromatic Graphs

A connected k-chromatic graph G is said to be double-critical if for all edges uv of G the graph G− u− v is (k− 2)-colourable. A longstanding conjecture of Erdős and Lovász states that the complete graphs are the only double-critical graphs. Kawarabayashi, Pedersen and Toft [Electron. J. Combin., 17(1): Research Paper 87, 2010] proved that every double-critical k-chromatic graph with k ≤ 7 contains a Kk minor. It remains unknown whether an arbitrary double-critical 8-chromatic graph contains a K8 minor, but in this paper we prove that any double-critical 8-chromatic contains a minor isomorphic to K8 with at most one edge missing. In addition, we observe that any double-critical 8-chromatic graph with minimum degree different from 10 and 11 contains a K8 minor.