The edge-minimal polyhedral maps of Euler characteristic \(-8\)
A polyhedral map \(P\) on a surface \(\mathbb M\) is called \(\{p,q\}\)-equivelar if each face of \(P\) is a \(p\)-gon and the degree of each vertex is \(q\). \textit{U. Brehm} [Topics in combinatorics and graph theory, 153-162 (1990; Zbl 0703.57008)] constructed a \(\{5,5\}\)-equivelar polyhedral map of Euler characteristic \(-8\). In this article the authors prove that for each \(m\geq 2\) there exists a selfdual \(\{5,5\}\)-equivelar polyhedral map on an orientable surface of Euler characteristic \(-4m\) and there exists a selfdual \(\{5,5\}\)-equivelar polyhedral map on a non-orientable surface of Euler characteristic \(-(4m+2)\). Moreover, it is proved that there is a unique \(\{5,5\}\)-equivelar polyhedral map on an orientable surface of Euler characteristic \(-8\).
scientific article; zbMATH DE number 1866014