The edge-minimal polyhedral maps of Euler characteristic - 8

Abstract

In [B], a $\{5, 5\}$-equivelar polyhedral map of Euler characteristic $-8$ was constructed. In this article we prove that $\{5, 5\}$-equivelar polyhedral map of Euler characteristic $-8$ is unique. As a consequence, we get that the minimum number of edges in a non-orientable polyhedral map of Euler characteristic $-8$ is $ > 40$. We have also constructed $\{5, 5\}$-equivelar polyhedral map of Euler characteristic $-2m$ for each $m ≥ 4$.