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Family sizes for complete multipartite graphs

Danielle Gregg, Thomas W. Mattman, Zachary Porat and George Todd

Vol. 15 (2022), No. 4, 669–686

The obstruction set for graphs with knotless embeddings is not known, but a recent paper of Goldberg, Mattman, and Naimi indicates that it is quite large. Almost all known obstructions fall into four triangle-Y families and they ask if there is an efficient way of finding or estimating the size of such graph families. Inspired by this question, we investigate the family size for complete multipartite graphs. Aside from three families that appear to grow exponentially, these families stabilize: after a certain point, increasing the number of vertices in a fixed part does not change family size.

spatial graphs, intrinsic knotting
Mathematical Subject Classification
Primary: 05C10
Secondary: 05C35, 57M15
Received: 2 September 2021
Revised: 21 December 2021
Accepted: 31 December 2021
Published: 7 January 2023

Communicated by Joel Foisy
Danielle Gregg
Department of Mathematics
Union College
Schenectady, NY
United States
Thomas W. Mattman
Department of Mathematics and Statistics
California State University
Chico, CA
United States
Zachary Porat
Department of Mathematics and Computer Science
Wesleyan University
Middletown, CT
United States
George Todd
Department of Mathematics
University of Dayton
Dayton, OH
United States