Simple graphs of order 12 and minimum degree 6 contain $K_6$ minors

Abstract

We prove that every simple graph of order 12 which has minimum degree 6 contains a $K_6$ minor, thus proving Jørgensen’s conjecture for graphs of order 12. In the process, we establish several lemmata linking the existence of $K_6$ minors for graphs to their size or degree sequence, by means of their clique sum structure. We also establish an upper bound for the order of graphs where the 6-connected condition is necessary for Jørgensen’s conjecture.