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

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.