Nonexistence of a Weakly Neighbourly Polyhedral Map of Type {6, 6}

Abstract

For the existence of an n-vertex polyhedral map of type {p, p}, it is known that n must be {\small $\geq (p-1)^2$} and equality holds if and only if K is weakly neighbourly. In 2002, Brehm et al. saw that there is a unique polyhedral map of type {\small $\{5, 5\}$} on 16 vertices. In 1990, Brehm constructed a polyhedral map of type {\small $\{6, 6\}$} with 26 vertices. In this article, we prove that there do not exist any polyhedral maps of type {\small $\{6, 6\}$} on 25 vertices. As a consequence, we show that the minimum number of edges in polyhedral maps of Euler characteristic -25 is {\small $>$} 75.