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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

• btained from G by a sequence of vertex deletions, edge

deletions, and edge contractions.

vertex deletion (5) edge deletion (23) 1 2 4 5 3 edge contraction (23) 1 4 5 2=3 1 2 3 4 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

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(Kn) = n.

1 2 3 4 5

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) 4 ⇒ n ≡ 0, 1(mod4)

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) ≤ ⌊ 6n

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 ≤ ⌊ 3n 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 4n vertices which do not contain a K2n+1 minor. (b) For n ≥ 0, there exist self-complementary graphs with 4n + 1 vertices which do not contain a K2n+2 minor.

Graph with 12 vertices and no K7 minor

Hadwiger numbers of self-complementary graphs The upper bound

K K E E X

q q q q r

Figure: Construction 1: a self-complementary graph with n = 4q + r vertices.

n = 4q + r r q h = ⌊ 3n

5 ⌋

20s 4s 4s 12s 20s + 1 4s + 1 4s 12s 20s + 4 4s 4s + 1 12s + 2 20s + 5 4s + 1 4s + 1 12s + 3 20s + 8 4s 4s + 2 12s + 4 20s + 9 4s + 1 4s + 2 12s + 5 20s + 13 4s + 1 4s + 3 12s + 7 20s + 16 4s + 4 4s + 3 12s + 9

Hadwiger numbers of self-complementary graphs The upper bound

Figure: A self-complementary graph with n = 12 vertices containing a K7

• minor. This minor can be obtained by contracting the five marked edges.

⌊12 + 1 2 ⌋ = 6 < 7 = ⌊3 · 12 5 ⌋

Hadwiger numbers of self-complementary graphs The upper bound

E2s+1 K2s+1 K2s+1 E2s+1 E E

2s+1 2s+1

K2s+1 K2s+1 T3 T4 T1 T2 Es Es Ks Ks a4 a2 a3 a1 Figure: Construction 2: A self-complementary graph on n = 20s + 12 vertices containing a K12s+7 minor. All the drawn edges represent complete bipartite graphs, and a1, a2, a3, and a4 are single vertices. All edges between T1, T2, T3, T4 and the two copies of Es and the two copies

• f Ks are also included in the graph.

Hadwiger numbers of self-complementary graphs The upper bound

K K E E X

4s+3 4s+4 4s+3 4s+3 4s+3

a

Figure: Construction 3: A self-complementary graph on n = 20s + 17 vertices containing a K12s+10 minor. The double dashed line marks that the highlighted vertex is adjacent to exactly half the vertices in X4s+4.

Hadwiger numbers of self-complementary graphs The induction step

Induction step n ⇒ n + 4 ⌊ n+1

2 ⌋ ≤ h ≤ ⌊ 3n 5 ⌋

• n+4+1

2

• −
• n+1

2

• = 2 and
• 3(n+4)

5

• −
• 3n

5

• ≤ 3

Xn

Figure: A self-complementary graph G with n + 4 vertices. Here Xn is self-complementary.

h(Xn) = 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

• uterplanar.

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!