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 K6 minor, thus proving Jørgensen’s conjecture for graphs of order 12. In the process, we establish several lemmata linking the existence of K6 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
Milestones
Received: 28 January 2020
Revised: 27 June 2020
Accepted: 28 June 2020
Published: 5 December 2020

Communicated by Ronald Gould
Authors
Ryan Odeneal
Department of Mathematics and Statistics
University of South Alabama
Mobile, AL
United States
Andrei Pavelescu
Department of Mathematics and Statistics
University of South Alabama
Mobile, AL
United States