We prove that every simple graph of order 12 which has minimum degree 6 contains a minor, thus proving Jørgensen’s conjecture for graphs of order 12. In the process, we establish several lemmata linking the existence of 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.
"Simple graphs of order 12 and minimum degree 6 contain $K_6$ minors." Involve 13 (5) 829 - 843, 2020. https://doi.org/10.2140/involve.2020.13.829