Graphs on 21 edges that are not 2-apex

Barsotti, Jamison; Mattman, Thomas W.

doi:10.2140/involve.2016.9.591

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Graphs on 21 edges that are not 2-apex

Jamison Barsotti and Thomas W. Mattman

Vol. 9 (2016), No. 4, 591–621

DOI: 10.2140/involve.2016.9.591

Abstract

We show that the 20-graph Heawood family, obtained by a combination of $\nabla Y$ and $Y \nabla$ moves on $K_{7}$ , is precisely the set of graphs of at most 21 edges that are minor-minimal with respect to the property “not $2$ -apex”. As a corollary, this gives a new proof that the 14 graphs obtained by $\nabla Y$ moves on $K_{7}$ are the minor-minimal intrinsically knotted graphs of 21 or fewer edges. Similarly, we argue that the seven-graph Petersen family, obtained from $K_{6}$ , is the set of graphs of at most 17 edges that are minor-minimal with respect to the property “not apex”.

Keywords

spatial graphs, intrinsic knotting, apex graphs, forbidden minors

Mathematical Subject Classification 2010

Primary: 05C10

Secondary: 57M15, 57M25

Milestones

Received: 12 January 2015

Revised: 23 June 2015

Accepted: 17 August 2015

Published: 6 July 2016

Communicated by Joel Foisy

Authors

Jamison Barsotti
Department of Mathematics University of California Santa Cruz, CA 95064 United States

Thomas W. Mattman
Department of Mathematics and Statistics California State University Chico, CA 95929-0525 United States

Vol. 9, No. 4, 2016