A new intrinsically knotted graph with 22 edges

Abstract

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell, and Michael showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh (and, independently, Barsotti and Mattman) proved there are exactly 14 intrinsically knotted graphs with 21 edges by showing that H-12 and C-14 are the only triangle-free intrinsically knotted graphs of size 21. Our current goal is to find the complete set of intrinsically knotted graphs with 22 edges. To this end, using the main argument in [9], we seek triangle-free intrinsically knotted graphs. In this paper we present a new intrinsically knotted graph with 22 edges, called M-11. We also show that there are exactly three triangle-free intrinsically knotted graphs of size 22 among graphs having at least two vertices with degree 5: cousins 94 and 110 of the E-9 + e family, and M11. Furthermore, there is no triangle-free intrinsically knotted graph with 22 edges that has a vertex with degree larger than 5. (c) 2017 Elsevier B.V. All rights reserved.