#### Vol. 13, No. 5, 2020

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

### Ryan Odeneal and Andrei Pavelescu

Vol. 13 (2020), No. 5, 829–843
##### 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.

##### Keywords
Jorgensen's conjecture, $K_6$ minors, minimum degree 6
##### Mathematical Subject Classification 2010
Primary: 05C10, 05C83