Properties of $8$-contraction-critical graphs with no $K_7$ minor

TL;DR

This paper investigates properties of 8-contraction-critical graphs lacking a K7 minor, showing such graphs have at most one vertex of degree 8, contributing to the broader goal of proving K7-minor-free graphs are 7-colorable.

Contribution

It establishes a structural property of 8-contraction-critical graphs without K7 minors, advancing the understanding towards Hadwiger's conjecture for K7.

Findings

Such graphs have at most one vertex of degree 8.

The result supports the conjecture that K7-minor-free graphs are 7-colorable.

Abstract

Motivated by the famous Hadwiger's Conjecture, we study the properties of $8$-contraction-critical graphs with no $K_7$ minor; we prove that every $8$-contraction-critical graph with no $K_7$ minor has at most one vertex of degree $8$, where a graph $G$ is $8$-contraction-critical if $G$ is not $7$-colorable but every proper minor of $G$ is $7$-colorable. This is one step in our effort to prove that every graph with no $K_7$ minor is $7$-colorable, which remains open.