Ashish Kumar Upadhyay

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 [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], 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 ann-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinctn-vertex weakly regular triangulations of the torus for eachn ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for eachm ≥ 2. For 12 ≤n ≤ 15, we have classified all then-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.

Contractible Hamiltonian cycles in triangulated surfaces

AbstractA triangulation of a surface is called q-equivelar if each of its vertices isincident with exactly qtriangles. In 1972 Altshuler had shown that an equivelartriangulation of torus has a Hamiltonian Circuit. Here we present a necessaryand sufficient condition for existence of a contractible Hamiltonian Cycle inequivelar triangulation of a surface.AMS classification: 57Q15, 57M20, 57N05.Keywords: Contractible Hamiltonian cycles, Proper Trees in Maps, EquivelarTriangulations. 1 Introduction A graph G := (V,E) is without loops and such that no more than one edge joins twovertices. A map on a surface S is an embedding of a graph G with finite number ofvertices such that the components of S \ G are topological 2-cells. Thus, the closureof a cell in S\G is a p−gonal disk, i.e. a 2-disk whose boundary is a p−gon for someinteger p ≥ 3.A map is called {p,q} equivelar if each vertex is incident with exactly q numbersof p-gons. If p = 3 then the map is called a q - equivelar triangulation or a degree- regular triangulation of type q. A map is called a

Degree-regular triangulations of the double-torus

Abstract 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 ℝ3. 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.

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.

Hamiltonian cycles in polyhedral maps

We present a necessary and sufficient condition for existence of a contractible, non-separating and non-contractible separating Hamiltonian cycle in the edge graph of polyhedral maps on surfaces. We also present algorithms to construct such cycles whenever it exists where one of them is linear time and another is exponential time algorithm.

Semi-equivelar maps on the torus and the Klein bottle with few vertices

Semi-equivelar maps are generalizations of maps on the surfaces of Archimedean solids to surfaces other than the 2-sphere. The well known 11 types of normal tilings of the plane suggest the possible types of semi-equivelar maps on the torus and the Klein bottle. In this article we classify (up to isomorphism) semi-equivelar maps on the torus and the Klein bottle with few vertices. c ©2017 Mathematical Institute Slovak Academy of Sciences

A note on edge-disjoint contractible Hamiltonian cycles in polyhedral maps

We present a necessary and sufficient condition for existence of edge-disjoint contractible Hamiltonian Cycles in the edge graph of polyhedral maps.