The complement of a Linklessly Embeddable Graph with at Least - PowerPoint PPT Presentation

The complement of a Linklessly Embeddable Graph with at Least Thirteen Vertices is Intrinsically Linked Andrei Bogdan Pavelescu Joint Work with Elena Pavelescu 31st Cumberland Conference University of Central Florida, Orlando, Florida May 18th 2019 Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

K n = the complete graph with n vertices. cG = the complement of G in K n . V ( cG ) = V ( G ), E ( cG ) = {{ i , j }| { i , j } / ∈ E ( G ) } . 1 1 5 5 2 2 4 4 3 3 Figure: Complementary graphs Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

For a graph G , a minor of G is any graph that can be obtained from G by a sequence of vertex deletions, edge deletions, and simple edge contractions. 1 1 vertex deletion (5) 5 4 2 2 4 3 3 1 1 edge deletion (23) 5 5 2 4 2 4 3 3 1 1 edge contraction 5 (23) 5 2 4 4 3 2=3 Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

Contracting edges produces subgraphs in the complement. 1 2 2 contraction(12) 5 3 3 5 4 4 complement complement 1 2 2 5 3 subgraph 5 3 4 4 Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

(N. Robertson and P. Seymour, ’83-’04) Every class of graphs closed under taking minors can be defined by a finite set of forbidden minors. A graph is a linear forest (disjoint union of paths) if and only if it does not have either of K 3 or K 1 , 3 as a minor. K 3 K 3,1 A graph is outerplanar if and only if it does not have either of K 4 or K 2 , 3 as a minor. K 4 K 3,2 Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

(K. Wagner, 1937) A graph is planar if and only if it does not have either of K 5 or K 3 , 3 as a minor. K 5 K 3,3 (V. Sivaraman, 2017) A graph does not have either of K 6 or K 4 , 3 as a minor if and only if ...? K 6 K 3,4 Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

A graph is called intrinsically linked ( IL ) if every one of its embeddings into R 3 contains a nontrivial link. A graph that is not intrinsically linked is called linklessly embeddable ( nIL ). (N. Robertson, P. Seymour, R. Thomas, 1993) A graph is nIL if and only if it does not have any of the Petersen family of graphs as a minor. Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

(J. Battle, F. Harary, Y. Kodama 1962) Every planar graph with nine points has a non planar complement. v 9 v 9 v v 10 10 v 7 v 7 v v v 3 5 1 v v v v v v 4 5 1 3 6 2 v v v v 8 4 2 6 v 8 (a) (b) Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

What is the minimal value of n such that any graph nIL graph of order n has an IL complement? (Campbell et al, 2008) Any graph on n vertices and at least 4 n − 9 edges contains a K 6 minor. n ( n − 1) / 2 ≥ 2(4 n − 9) ⇒ n 2 − 17 n + 36 ≥ 0 ⇒ n ≥ 15. Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

16 ? 10 Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

Y.C. de Verdi` ere, 1987 Let R ( n ) denote the space of real symmetric n × n matrices. If G = ( V , E ) is a graph of order n , then µ ( G ) is the largest corank of any matrix M = ( M i , j ) ∈ R ( n ) such that: for all i , j with i � = j , M i , j < 0 if i and j are adjacent, and M i , j = 0 if i and j are not adjacent; M has exactly one negative eigenvalue, of multiplicity 1; There is no nonzero matrix X ∈ R ( n ) such that MX = 0 and such that X i , j = 0 whenever i = j or M i , j � = 0. Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

If H is a minor of G , then µ ( H ) ≤ µ ( G ); µ ( G ) ≤ 1 if and only if G is a disjoint union of paths; µ ( G ) ≤ 2 if and only if G is outer planar; µ ( G ) ≤ 3 if and only if G is planar; µ ( G ) ≤ 4 if and only if G is nIL; µ ( G ) ≤ 5 if and only if G is ?; Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

µ ( G ) ≤ µ ( G − v ) + 1 , ∀ v ∈ V ( G ); v Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

If G is a disjoint union of paths, then µ ( cG ) ≥ n − 3; If G is outer planar, then µ ( cG ) ≥ n − 4; If G is planar, then µ ( cG ) ≥ n − 5; If G is nIL, then µ ( cG ) ≥ ... ? (A. Kotlov, L. Lov´ asz, S. Vempala, 1996) µ ( G ) + µ ( cG ) ≥ n − 2. Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

Lemma Consider a graph G with n ≥ 10 vertices. If G is planar, then its complement cG is IL. Lemma For a graph G with n vertices, n ≥ 11 , if there exists a vertex of G whose degree is at least 10, either G or its complement is intrinsically linked. Lemma Assume G is a graph with at least 12 vertices. If an edge contraction in G creates a vertex of degree at least 10, then either G or cG is intrinsically linked. Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

Lemma For n ≥ 12 , if maxdeg ( G ) ≥ 9 then G or cG is intrinsically linked. v 2 v 10 v 11 v 12 G v n v 1 v 11 v 1 v 2 v 10 v 12 cG v n Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

Theorem (A. Pavelescu, E. Pavelescu, 2019 ) Let G denote a simple graph with 13 vertices. Then either G or cG is intrinsically linked. v 2 v 9 v 10 v 11 G v 12 v 13 v 1 v 10 v 1 v 2 v 9 v 11 cG v 12 v 13 Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

Theorem (A. Pavelescu, E. Pavelescu, 2019 ) Let G denote a simple graph with 13 vertices. Then either G or cG is intrinsically linked. v 2 v 3 v 9 v 10 v 11 G v 12 v 13 v 1 v 10 v 1 v 2 v 9 v 11 cG v 12 v 13 Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL

0 12 1 11 2 10 3 4 9 5 8 6 7 Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL Figure: Paley graph with thirteen vertices. Contracting the highlighted

Abdee, abdee, abdee, that’s all folks! Andrei Pavelescu The Complement of a nIL graphs with 13 vertices is IL