# A note on the existence of $\mathbf{\{k, k\}}$-equivelar polyhedral maps

Datta, Basudeb (2005) A note on the existence of $\mathbf{\{k, k\}}$-equivelar polyhedral maps. In: Contributions to Algebra and Geometry / Beitrage zur Algebra und Geometrie, 46 (2). pp. 537-544.

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## Abstract

A polyhedral map is called $\{p,q\}$-equivelar if each face has $p$ edges and each vertex belongs to $q$ faces. In , 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 \cite{dn1} that there exist infinitely many $\{4, 4\}$- and $\{3, 6\}$-equivelar polyhedral maps. In \cite{b}, 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 $k \geq 5$. Also vertex-minimal non-singular $\{p,p\}$-pattern are constructed for all odd primes $p$.

Item Type: Journal Article Copyright of this article belogns to Heldermann Verlag. Polyhedral maps;equivelar maps;non-singular patterns. Division of Physical & Mathematical Sciences > Mathematics 22 Jul 2008 27 Aug 2008 13:37 http://eprints.iisc.ernet.in/id/eprint/15159