Degree-regular triangulations of torus and Klein bottle

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dc.creatorDatta, Basudeb
dc.creatorUpadhyay, Ashish Kumar
dc.date2004-03-25
dc.date2005-07-06
dc.date.accessioned2026-07-07T05:06:44Z
dc.date.available2026-07-07T05:06:44Z
dc.descriptionA triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In 1999, Lutz has classified all the weakly regular triangulations on at most 15 vertices. In 2001, Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices. In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists an $n$-vertex degree-regular triangulation of the Klein bottle if and only if $n$ is a composite number $\geq 9$. We have constructed two distinct $n$-vertex weakly regular triangulations of the torus for each $n \geq 12$ and a $(4m + 2)$-vertex weakly regular triangulation of the Klein bottle for each $m \geq 2$. For $12 \leq n \leq 15$, we have classified all the $n$-vertex degree-regular triangulations of the torus and the Klein bottle. There are exactly 19 such triangulations, 12 of which are triangulations of the torus and remaining 7 are triangulations of the Klein bottle. Among the last 7, only one is weakly regular.
dc.descriptionRevised version, 26 pages, To appear in Proceedings of Indian Academy of Sciences (Math. Sci.)
dc.identifierhttps://arxiv.org/abs/math/0403433
dc.identifierhttp://arxiv.org/abs/math/0403433
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/70588
dc.subjectGeometric Topology
dc.subjectAlgebraic Topology
dc.subject57Q15; 57M20; 57N05
dc.titleDegree-regular triangulations of torus and Klein bottle
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