Double-critical graphs and complete minors

TL;DR

This paper advances the understanding of double-critical graphs by proving that for k=6 and 7, any non-complete double-critical k-chromatic graph must contain a complete minor, supporting the conjecture that only complete graphs are double-critical.

Contribution

It proves that for k=6 and 7, non-complete double-critical graphs are 6-connected and contain a complete minor, providing new evidence for the conjecture that only complete graphs are double-critical.

Findings

Non-complete double-critical graphs for k=6,7 are 6-connected.

Such graphs contain a complete minor K_k.

Supports the conjecture that only complete graphs are double-critical.

Abstract

A connected $k$-chromatic graph $G$ is double-critical if for all edges $uv$ of $G$ the graph $G - u - v$ is $(k-2)$-colourable. The only known double-critical $k$-chromatic graph is the complete $k$-graph $K_k$. The conjecture that there are no other double-critical graphs is a special case of a conjecture from 1966, due to Erd\H{o}s and Lov\'asz. The conjecture has been verified for $k \leq 5$. We prove for $k=6$ and $k=7$ that any non-complete double-critical $k$-chromatic graph is 6-connected and has $K_k$ as a minor.