Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 46, No. 2, pp. 537-544 (2005)

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

Basudeb Datta

Department of Mathematics, Indian Institute of Science, Bangalore 560\,012, India; e-mail:

Abstract: A polyhedral map is called $\{p,q\}$-equivelar if each face has $p$ edges and each vertex belongs to $q$ faces. In \cite{msw2}, 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 $p$.

Keywords: polyhedral maps, equivelar maps, non-singular patterns

Classification (MSC2000): 52B70, 51M20, 57M20

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Electronic version published on: 18 Oct 2005. This page was last modified: 29 Dec 2008.

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