Bipartite intrinsically knotted graphs with 23 edges

Hyoungjun Kim, Thomas Mattman, Seungsang Oh

Abstract: A graph is intrinsically knotted if every embedding contains a nontrivially knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that there are exactly 14 intrinsically knotted graphs with 21 edges, in which the Heawood graph is the only bipartite graph. The authors showed that there are exactly two graphs with at most 22 edges that are minor minimal bipartite intrinsically knotted: the Heawood graph and Cousin 110 of the E9+e<math><msub is="true"><mrow is="true"><mi is="true">E</mi></mrow><mrow is="true"><mn is="true">9</mn></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mi is="true">e</mi></math> family. In this paper we show that there are exactly six bipartite intrinsically knotted graphs with 23 edges so that every vertex has degree 3 or more. Four among them contain the Heawood graph and the other two contain Cousin 110 of the E9+e<math><msub is="true"><mrow is="true"><mi is="true">E</mi></mrow><mrow is="true"><mn is="true">9</mn></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mi is="true">e</mi></math> family. Consequently, there is no minor minimal intrinsically knotted graph with 23 edges that is bipartite.