Many, many more intrinsically knotted graphs

Goldberg, Noam; Mattman, Thomas W.; Naimi, Ramin

doi:10.2140/agt.2014.14.1801

Cited by 34 publications

(79 citation statements)

References 14 publications

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“…Let A n denote the set of vertices in A of degree n = 3, 4, 5 and [A] = [|A 5 |, |A 4 |, |A 3 |] and similarly for B. [2,0,4], [1,2,3], [0, 4, 2], or [0, 1,6]. Without loss of generality, we may assume that |A 5 | ≥ |B 5 |, and if…”

Section: Propositionmentioning

confidence: 99%

“…The K 3, 3, 1, 1 family consists of 58 graphs, all of which are known to be minor minimal intrinsically knotted by Goldberg et al. . The graph

E_{9} + e

shown in Figure is obtained from N 9 by adding a new edge e and has a family of 110 graphs.…”

Section: Introductionmentioning

confidence: 99%

“…The graph

E_{9} + e

shown in Figure is obtained from N 9 by adding a new edge e and has a family of 110 graphs. All of these graphs are intrinsically knotted and exactly 33 are minor minimal intrinsically knotted . The 168 graphs that make up the K 3, 3, 1, 1 family and

E_{9} + e

family together, are all intrinsically knotted graphs with 22 edges.…”

Section: Introductionmentioning

confidence: 99%

“…Goldberg et al. showed that all graphs in this family are intrinsically knotted. However, Cousin 89 is formed by adding an edge to the Heawood graph and is not minor minimal.…”

Section: Introductionmentioning

confidence: 99%

“…The set of all cousins of G is In this article, we concentrate on intrinsically knotted graphs with 22 edges. The K 3,3,1,1 family consists of 58 graphs, all of which are known to be minor minimal intrinsically knotted by Goldberg et al [6]. The graph E 9 + e shown in Figure 3 is obtained from N 9 by adding a new edge e and has a family of 110 graphs.…”

Section: Introductionmentioning

confidence: 99%

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Bipartite Intrinsically Knotted Graphs with 22 Edges

Kim¹,

Mattman

2016

Journal of Graph Theory

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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.

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Section: Propositionmentioning

confidence: 99%

“…The K 3, 3, 1, 1 family consists of 58 graphs, all of which are known to be minor minimal intrinsically knotted by Goldberg et al. . The graph

E_{9} + e

shown in Figure is obtained from N 9 by adding a new edge e and has a family of 110 graphs.…”

Section: Introductionmentioning

confidence: 99%

“…The graph

E_{9} + e

E_{9} + e

family together, are all intrinsically knotted graphs with 22 edges.…”

Section: Introductionmentioning

confidence: 99%

“…Goldberg et al. showed that all graphs in this family are intrinsically knotted. However, Cousin 89 is formed by adding an edge to the Heawood graph and is not minor minimal.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bipartite Intrinsically Knotted Graphs with 22 Edges

Kim¹,

Mattman

2016

Journal of Graph Theory

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Linearly free graphs

Huh

Lee

2021

Journal of Graph Theory

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In this paper we are interested in an intrinsic property of graphs which is derived from their embeddings into the Euclidean 3‐space R 3. An embedding of a graph into R 3 is said to be linear, if it sends every edge to be a line segment. And we say that an embedding f of a graph G into R 3 is free, if π 1 ( R 3 − f ( G ) ) is a free group. Lastly a simple connected graph is said to be linearly free if every its linear embedding is free. In the 1980s it was proved that every complete graph is linearly free, by Nicholson. In this paper, we develop Nicholson's arguments into a general notion, and establish a sufficient condition for a linear embedding to be free. As an application of the condition we give a partial answer for a question: How much can the complete graph K n be enlarged so that the linear‐freeness is preserved and the clique number does not increase? And an example supporting our answer is provided. As the second application it is shown that a simple connected graph of minimal valency at least 3 is linearly free, if it has less than eight vertices. The conditional inequality is strict, because we found a graph with eight vertices which are not linearly free. It is also proved that for n , m ≤ 6 the complete bipartite graph K n , m is linearly free.

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Forbidden Minors: Finding the Finite Few

Mattman¹

2017

Foundations for Undergraduate Research in Mathematics

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Abstract. The Graph Minor Theorem of Robertson and Seymour asserts that any graph property, whatsoever, is determined by an associated finite lists of graphs. We view this as an impressive generalization of Kuratowski's theorem, which chracterizes planarity in terms of two forbidden subgraphs, K 5 and K 3,3 . Robertson and Seymour's result empowers students to devise their own Kuratowski type theorems; we propose several undergraduate research projects with that goal. As an explicit example, we determine the seven forbidden minors for a property we call strongly almost-planar (SAP). A graph is SAP if, for any edge e, both deletion and contraction of e result in planar graphs.

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Many, many more intrinsically knotted graphs

Abstract: We list more than 200 new examples of minor minimal intrinsically knotted graphs and describe many more that are intrinsically knotted and likely minor minimal.Comment: 19 pages, 16 figures, Appendi

Cited by 34 publications

References 14 publications

Bipartite Intrinsically Knotted Graphs with 22 Edges

Bipartite Intrinsically Knotted Graphs with 22 Edges

Linearly free graphs

Forbidden Minors: Finding the Finite Few

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