Hadwiger numbers of self-complementary graphs Elena Pavelescu - PowerPoint PPT Presentation

Hadwiger numbers of self-complementary graphs Hadwiger numbers of self-complementary graphs Elena Pavelescu University of South Alabama joint work with Andrei Pavelescu 31st Cumberland Conference University of Central Florida, 18-19 May 2019

Hadwiger numbers of self-complementary graphs Definitions, Main result 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 edge contractions. 1 1 vertex deletion (5) 5 4 2 4 2 3 3 1 1 edge deletion (23) 5 5 4 2 2 4 3 3 1 1 edge contraction 5 (23) 5 2 4 4 3 2=3

Hadwiger numbers of self-complementary graphs Definitions, Main result The Hadwiger number of a graph G , h ( G ), is the size of the largest complete minor of G . h ( K n ) = n . 1 5 2 4 3 Figure: h ( G ) = 4

Hadwiger numbers of self-complementary graphs Definitions, Main result A self-complementary graph G is a graph which is isomorphic to its complement, that is, there exists a graph isomorphism ρ : V ( G ) → V ( cG ) If G is a self-complementary graph with n vertices, | E ( G ) | = n ( n − 1) ⇒ n ≡ 0 , 1( mod 4) 4 Figure: Self-complementary graphs of order 1, 4 and 5

Hadwiger numbers of self-complementary graphs Definitions, Main result Figure: Self-complementary graphs of order 8

Hadwiger numbers of self-complementary graphs Definitions, Main result (M. Stiebitz, ’92) h ( G ) + h ( cG ) ≤ ⌊ 6 n 5 ⌋ ; (Girse-Gillman ’88, Rao-Sahoo ’14, Pavelescu-P ’18) h ( G ) ≥ n + 1 2 Theorem (Pavelescu-P. ’18) For all n ≡ 0 , 1( mod 4 ) and for all ⌊ n +1 2 ⌋ ≤ h ≤ ⌊ 3 n 5 ⌋ , there exists a self-complementary graph G with n vertices whose Hadwiger number is h.

Hadwiger numbers of self-complementary graphs Definitions, Main result Example (a) For n ≥ 1, there exist self-complementary graphs with 4 n vertices which do not contain a K 2 n +1 minor. (b) For n ≥ 0, there exist self-complementary graphs with 4 n + 1 vertices which do not contain a K 2 n +2 minor. Graph with 12 vertices and no K 7 minor

Hadwiger numbers of self-complementary graphs The upper bound X r E K E K q q q q Figure: Construction 1 : a self-complementary graph with n = 4 q + r vertices. h = ⌊ 3 n n = 4 q + r 5 ⌋ r q 20 s 4 s 4 s 12 s 20 s + 1 4 s + 1 4 s 12 s 20 s + 4 4 s 4 s + 1 12 s + 2 20 s + 5 4 s + 1 4 s + 1 12 s + 3 20 s + 8 4 s 4 s + 2 12 s + 4 20 s + 9 4 s + 1 4 s + 2 12 s + 5 20 s + 13 4 s + 1 4 s + 3 12 s + 7 20 s + 16 4 s + 4 4 s + 3 12 s + 9

Hadwiger numbers of self-complementary graphs The upper bound Figure: A self-complementary graph with n = 12 vertices containing a K 7 minor. This minor can be obtained by contracting the five marked edges. ⌊ 12 + 1 ⌋ = 6 < 7 = ⌊ 3 · 12 ⌋ 2 5

Hadwiger numbers of self-complementary graphs The upper bound T 3 T 4 E 2s+1 E 2s+1 a 3 K 2s+1 K 2s+1 a 4 E s K s K s E s K 2s+1 a 2 K 2s+1 a 1 E 2s+1 E 2s+1 T 1 T 2 Figure: Construction 2 : A self-complementary graph on n = 20 s + 12 vertices containing a K 12 s +7 minor. All the drawn edges represent complete bipartite graphs, and a 1 , a 2 , a 3 , and a 4 are single vertices. All edges between T 1 , T 2 , T 3 , T 4 and the two copies of E s and the two copies of K s are also included in the graph.

Hadwiger numbers of self-complementary graphs The upper bound X 4s+4 a E K K E 4s+3 4s+3 4s+3 4s+3 Figure: Construction 3: A self-complementary graph on n = 20 s + 17 vertices containing a K 12 s +10 minor. The double dashed line marks that the highlighted vertex is adjacent to exactly half the vertices in X 4 s +4 .

Hadwiger numbers of self-complementary graphs The induction step Induction step n ⇒ n + 4 ⌊ n +1 2 ⌋ ≤ h ≤ ⌊ 3 n 5 ⌋ � � � � � � � � 3( n +4) n +4+1 n +1 3 n − = 2 and − ≤ 3 2 2 5 5 X n Figure: A self-complementary graph G with n + 4 vertices. Here X n is self-complementary. h ( X n ) = h ⇒ h ( G ) = h + 2

Hadwiger numbers of self-complementary graphs Topological properties of graphs Any self-complementary graph on n ≥ 8 vertices in not outerplanar. Figure: Self-complementary graphs on 8 vertices Any self-complementary graph on n ≥ 9 vertices in not planar, but it is linklessly embeddable.

Hadwiger numbers of self-complementary graphs Topological properties of graphs Any self-complementary graph on n ≥ 12 vertices is intrinsically linked. a b (a) (b) Figure: (a) A self-complementary graph on 12 vertices is obtained by pairwise connecting all filled vertices, (b) Removing vertices a and b gives a planar subgraph, so the graph is not intrinsically knotted. Any self-complementary graph on n ≥ 13 vertices in intrinsically knotted.

Hadwiger numbers of self-complementary graphs Topological properties of graphs Thank you!