Properties of 8-contraction-critical graphs with no K7 minor

Motivated by the famous Hadwiger’s Conjecture, we study the properties of 8-contraction-critical graphs with no K7 minor. In particular, we prove that every 8-contraction-critical graph with no K7 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 K7 minor is 7-colorable, which remains open.