Intrinsically Knotted Graphs Have at Least 21 Edges

J. Knot Theory Ramifications

Oh³

2018

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael, and, independently, Mattman showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that K7 and the 13 graphs obtained from K7 by ∇Y moves are the only intrinsically knotted graphs with 21 edges. Also Kim, Lee, Lee, Mattman and Oh showed that there are exactly three triangle-free intrinsically knotted graphs with 22 edges having at least two vertices of degree 5. Furthermore, there is no triangle-free intrinsically knotted graph with 22 edges that has a vertex with degree larger than 5. In this paper we show that there are exactly five trianglefree intrinsically knotted graphs with 22 edges having exactly one degree 5 vertex. These are Cousin 29 of the K3,3,1,1 family, Cousins 97 and 99 of the E9 + e family and two others that were previously unknown.

Section: Introductionmentioning

confidence: 99%

More intrinsically knotted graphs with 22 edges and the restoring method

Kim¹,

J. Knot Theory Ramifications

Oh³

2018

“…Johnson et al. and, independently, the second author showed that intrinsically knotted graphs have at least 21 edges. Hanaki et al.…”

Section: Introductionmentioning

confidence: 99%

“…The only bipartite graph in this set is C 14 , the Heawood graph. As graphs of 20 edges are not intrinsically knotted , C 14 is minor minimal for intrinsic knotting and, so, also for bipartite intrinsically knotted. By Theorem , if

∥ G ∥ = 22

, G must be one of the two cousins in the

E_{9} + e

family.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Bipartite Intrinsically Knotted Graphs with 22 Edges

Kim¹,

Journal of Graph Theory

2016

Abstract. A graph is intrinsically knotted if every embedding contains a knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that the KS graphs, K7 and the 13 graphs obtained from K7 by ∇Y moves, are the only minor minimal intrinsically knotted graphs with 21 edges [1,8,10,11]. This set includes exactly one bipartite graph, the Heawood graph.In this paper we classify the intrinsically knotted bipartite graphs with at most 22 edges. Previously known examples of intrinsically knotted graphs of size 22 were those with KS graph minor and the 168 graphs in the K3,3,1,1 and E9 + e families. Among these, the only bipartite example with no Heawood subgraph is Cousin 110 of the E9 + e family. We show that, in fact, this is a complete listing. That is, there are exactly two graphs of size at most 22 that are minor minimal bipartite intrinsically knotted: the Heawood graph and Cousin 110.

“…As part of their proof that intrinsic knotting requires 21 edges, Johnson, Kidwell and Michael [5] showed that every triangle-free graph on 20 or fewer edges is 2-apex and, therefore, not knotted. In the current paper, we show: This suggests the following: Question 1.5 Is every non-IK graph 2-apex?…”

Section: Introductionmentioning

confidence: 99%

Graphs of 20 edges are 2–apex, hence unknotted

2011

Algebr. Geom. Topol.

Graphs of 20 edges are 2-apex, hence unknotted THOMAS W MATTMANA graph is 2-apex if it is planar after the deletion of at most two vertices. Such graphs are not intrinsically knotted, IK. We investigate the converse, does not IK imply 2-apex? We determine the simplest possible counterexample, a graph on nine vertices and 21 edges that is neither IK nor 2-apex. In the process, we show that every graph of 20 or fewer edges is 2-apex. This provides a new proof that an IK graph must have at least 21 edges. We also classify IK graphs on nine vertices and 21 edges and find no new examples of minor minimal IK graphs in this set. 05C10; 57M15 IntroductionWe say that a graph is intrinsically knotted or IK if every tame embedding of the graph in R 3 contains a nontrivially knotted cycle. Blain et al [1] and Ozawa and Tsutsumi [9] independently discovered an important criterion for intrinsic knotting. Let H K 2 denote the join of the graph H and the complete graph on two vertices, K 2 . Proposition 1.1 [1; 9] A graph of the form H K 2 is IK if and only if H is nonplanar.A graph is called l -apex if it becomes planar after the deletion of at most l vertices (and their edges). The proposition shows that 2-apex graphs are not IK.It's known that many non-IK graphs are 2-apex. As part of their proof that intrinsic knotting requires 21 edges, Johnson, Kidwell and Michael [5] showed that every triangle-free graph on 20 or fewer edges is 2-apex and, therefore, not knotted. In the current paper, we show: This suggests the following: Question 1.5 Is every non-IK graph 2-apex?We answer the question in the negative by giving an example of a graph, E 9 , having nine vertices and 21 edges that is neither IK nor 2-apex. (We thank Ramin Naimi [8] for providing an unknotted embedding of E 9 , which appears as Figure 8 in Section 3. This graph is called N 9 in [4].) Further, we show that no graph on fewer than 21 edges, no graph on fewer than nine vertices, and no other graph on 21 edges and nine vertices has this property. In this sense, E 9 is the simplest possible counterexample to our Question.The notation E 9 is meant to suggest that this graph is a "cousin" to the set of 14 graphs derived from K 7 by triangle-Y moves (see Kohara and Suzuki [6]). Indeed, E 9 arises from a Y-triangle move on the graph F 10 in the K 7 family. Although intrinsic knotting is preserved under triangle-Y moves by Motwani, Raghunathan and Saran [7], it is not, in general, preserved under Y-triangle moves. For example, although F 10 is derived from K 7 by triangle-Y moves and, therefore, intrinsically knotted, the graph E 9 , obtained by a Y-triangle move on F 10 , has an unknotted embedding.Our analysis includes a classification of IK and 2-apex graphs on nine vertices and at most 21 edges. Such a graph is 2-apex unless it is E 9 , or, up to addition of degree zero vertices, one of four graphs derived from K 7 by triangle-Y moves [6]. (Here jGj denotes the number of vertices in the graph G and kGk is the number of edges.) Proposition 1.6 Let G be a graph with ...