A note on the existence of {k, k}-equivelar polyhedral maps

Abstract

A polyhedral map is called $\{p,q\}$-equivelar if each face has $p$ edges and each vertex belongs to $q$ faces. In 1983, it was shown that there exist infinitely many geometrically realizable $\{p,q\}$-equivelar polyhedral maps if $q>p=4$, $p>q=4$ or $q-3>p=3$. It was shown in 2001 that there exist infinitely many $\{4,4\}$- and $\{3,6\}$-equivelar polyhedral maps. In 1990, it was shown that $\{5,5\}$- and $\{6,6\}$-equivelar polyhedral maps exist. In this note, examples are constructed, to show that infinitely many self dual $\{k,k\}$-equivelar polyhedral maps exist for each $k\ge 5$. Also vertex-minimal non-singular $\{p,p\}$-pattern are constructed for all odd primes $p$.

7 pages. To appear in `Contributions to Algebra and Geometry'