Hadwiger Numbers of 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. It is known that for any \(n\equiv 0,1 (\text {mod 4})\) and any self-complementary graph G with n vertices, \( \lfloor (n+1)/2 \rfloor \le h(G) \le \lfloor 3n/5\rfloor \). In this article, we revisit the proof of the lower bound, and we prove that for all \(n\equiv 0,1 (\text {mod 4})\) and \( \lfloor (n+1)/2 \rfloor \le h \le \lfloor 3n/5\rfloor \), there exists a self-complementary graph G with n vertices whose Hadwiger number is h. We visit topological properties of self-complementary graphs.