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

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

2005-06-30

arxiv.org/abs/math/0506618v1

Mathematics

Geometric Topology

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

Scientific paper

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 $k \geq 5$. Also vertex-minimal non-singular $\{p, p\}$-pattern are constructed for all odd primes $p$.

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