Vol. 9, No. 4, 2016

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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

We show that the 20-graph Heawood family, obtained by a combination of Y and Y moves on K7, 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 Y moves on K7 are the minor-minimal intrinsically knotted graphs of 21 or fewer edges. Similarly, we argue that the seven-graph Petersen family, obtained from K6, is the set of graphs of at most 17 edges that are minor-minimal with respect to the property “not apex”.

spatial graphs, intrinsic knotting, apex graphs, forbidden minors
Mathematical Subject Classification 2010
Primary: 05C10
Secondary: 57M15, 57M25
Received: 12 January 2015
Revised: 23 June 2015
Accepted: 17 August 2015
Published: 6 July 2016

Communicated by Joel Foisy
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