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 ≥ (p − 1)² and equality holds if and only if K is weakly neighbourly. We have seen in [Brehm et al. 02] that there is a unique polyhedral map of type {5,5} on 16 vertices. In [Brehm 90], Brehm constructed a polyhedral map of type {6,6} with 26 vertices. In this article, we prove that there do not exist any polyhedral maps of type {6,6} on 25 vertices. As a consequence, we show that the minimum number of edges in polyhedral maps of Euler characteristic −25 is > 75.

Keywords:

• Polyhedral maps
• polyhedral 2-manifolds
• regular graph design