Simple graphs of order 12 and minimum degree 6 contain K6 minors

Odeneal, Ryan; Pavelescu, Andrei

doi:10.2140/involve.2020.13.829

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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

DOI: 10.2140/involve.2020.13.829

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

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

Vol. 13, No. 5, 2020