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Complete minors in complements of nonseparating planar graphs

Leonard Fowler, Gregory Li and Andrei Pavelescu

Vol. 16 (2023), No. 3, 505–518
Abstract

We prove that the complement of any nonseparating planar graph of order 2n 3 contains a Kn minor and argue that the order 2n 3 is lowest possible with this property. To illustrate the necessity of the nonseparating hypothesis, we give an example of a planar graph of order 11 whose complement does not contain a K7 minor. We argue that the complements of planar graphs of order 11 are intrinsically knotted. We compute the Hadwiger numbers of complements of wheel graphs.

Keywords
graph minors, intrinsically knotted, nonseparating planar graphs, Hadwiger number
Mathematical Subject Classification
Primary: 05C10
Secondary: 57M15
Milestones
Received: 2 May 2022
Revised: 15 July 2022
Accepted: 22 July 2022
Published: 10 August 2023

Communicated by Joel Foisy
Authors
Leonard Fowler
Rensselaer Polytechnic Institute
Troy, NY
United States
Gregory Li
Harvard University
Cambridge, MA
United States
Andrei Pavelescu
Department of Mathematics and Statistics
University of South Alabama
Mobile, AL
United States