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 $k \leq 7$ contains a $K_k$ minor. It remains unknown whether an arbitrary double-critical $8$-chromatic graph contains a $K_8$ minor, but in this paper we prove that any double-critical $8$-chromatic contains a minor isomorphic to $K_8$ 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 $K_8$ minor.