Complete and almost complete minors in double-critical 8-chromatic graphs

TL;DR

This paper proves that double-critical 8-chromatic graphs contain a nearly complete minor, specifically a $K_8^-$ minor, and identifies conditions under which they contain a complete $K_8$ minor, advancing understanding of graph minors in critical graphs.

Contribution

It establishes that double-critical 8-chromatic graphs always contain a $K_8^-$ minor, and shows that certain degree conditions guarantee a $K_8$ minor, extending previous results for lower chromatic numbers.

Findings

Double-critical 8-chromatic graphs contain a $K_8^-$ minor.

Graphs with minimum degree not 10 or 11 contain a $K_8$ minor.

Supports the conjecture that such graphs contain large minors.

Abstract

A connected $k$-chromatic graph $G$ is said to be {\it double-critical} if for all edges $uv$ of $G$ the graph $G - u - v$ is $(k-2)$-colourable. A longstanding conjecture of Erd\H{o}s and Lov\'asz states that the complete graphs are the only double-critical graphs. Kawarabayashi, Pedersen and Toft [\it{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 $K_8^-$ minor; here $K_8^-$ denotes the complete $8$-graph with 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.