Hadwiger numbers of self-complementary graphs

TL;DR

This paper investigates the Hadwiger numbers of self-complementary graphs, establishing the existence of such graphs with specific Hadwiger numbers for certain vertex counts, advancing understanding of their structural properties.

Contribution

It proves the existence of self-complementary graphs with prescribed Hadwiger numbers within a certain range for all relevant vertex counts.

Findings

Existence of self-complementary graphs with given Hadwiger numbers for n ≡ 0,1 mod 4.

Range of Hadwiger numbers for which such graphs exist.

Extension of known results on graph minors and self-complementary graphs.

Abstract

The Hadwiger number of a graph $G$, denoted by $h(G)$, is the order of the largest complete minor of $G$. A graph is said to be self-complementary if it is isomorphic to its complement. We prove that for all $n\equiv 0,1 (\text{mod 4})$ and for all $ \lfloor \dfrac{n+1}{2} \rfloor \le h \le \lfloor \dfrac{3n}{5}\rfloor $, there exists a self-complementary graph $G$ with $n$ vertices whose Hadwiger number is $h$.