132 research outputs found
Some Centrally Symmetric Manifolds
We show existence of centrally symmetric maps on surfaces all of whose faces
are quadrangles and pentagons for each orientable genus . We also
show existence of centrally symmetric maps on surfaces all of whose faces are
hexagons for each orientable genus , . We enumerate
centrally symmetric triangulated manifolds of dimensions 2 and 3 with few
vertices.Comment: 18 page
Degree-regular triangulations of the double-torus
A connected combinatorial 2-manifold is called degree-regular if each of its
vertices have the same degree. A connected combinatorial 2-manifold is called
weakly regular if it has a vertex-transitive automorphism group. Clearly, a
weakly regular combinatorial 2-manifold is degree-regular and a degree-regular
combinatorial 2-manifold of Euler characteristic - 2 must contain 12 vertices.
In 1982, McMullen et al. constructed a 12-vertex geometrically realized
triangulation of the double-torus in \RR^3. As an abstract simplicial
complex, this triangulation is a weakly regular combinatorial 2-manifold. In
1999, Lutz showed that there are exactly three weakly regular orientable
combinatorial 2-manifolds of Euler characteristic - 2. In this article, we
classify all the orientable degree-regular combinatorial 2-manifolds of Euler
characteristic - 2. There are exactly six such combinatorial 2-manifolds. This
classifies all the orientable equivelar polyhedral maps of Euler characteristic
- 2.Comment: 13 pages. To appear in `Forum Mathematicum
Contractible Hamiltonian Cycles in Polyhedral Maps
We present a necessary and sufficient condition for existence of a
contractible Hamiltonian Cycle in the edge graph of equivelar maps on surfaces.
We also present an algorithm to construct such cycles. This is further
generalized and shown to hold for more general maps.Comment: 9 pages, 1 figur
Degree-regular triangulations of torus and Klein bottle
A 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 -vertex
degree-regular triangulation of the Klein bottle if and only if is a
composite number . We have constructed two distinct -vertex weakly
regular triangulations of the torus for each and a -vertex weakly regular triangulation of the Klein bottle for each . For , we have classified all the -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.Comment: Revised version, 26 pages, To appear in Proceedings of Indian Academy
of Sciences (Math. Sci.
Hamiltonian Cycles in Polyhedral Maps
We present a necessary and sufficient condition for existence of a
contractible, non-separating and noncontractible separating Hamiltonian cycle
in the edge graph of polyhedral maps on surfaces. In particular, we show the
existence of contractible Hamiltonian cycle in equivelar triangulated maps. We
also present an algorithm to construct such cycles whenever it exists.Comment: 14 page
Corrigendum to ``"On the enumeration of a class of toroidal graphs" [Contrib. Discrete Math. 13 (2018), no. 1, 79-119]
Corrigendum to ``On the enumeration of a class of toroidal graphs [Contrib. Discrete Math. 13 (2018), no. 1, 79-119
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