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

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

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 $g \geq 0$. We also show existence of centrally symmetric maps on surfaces all of whose faces are hexagons for each orientable genus $g = 2k-1$, $k\in \mathbb{N}$. We enumerate centrally symmetric triangulated manifolds of dimensions 2 and 3 with few vertices.Comment: 18 page

Semi-equivelar toroidal maps and their vertex covers

If the face\mbox{-}cycles at all the vertices in a map are of same type then the map is called semi\mbox{-}equivelar. A map is called minimal if the number of vertices is minimal. We know the bounds of number of vertex orbits of semi-equivelar toroidal maps. These bounds are sharp. Datta \cite{BD2020} has proved that every semi-equivelar toroidal map has a vertex-transitive cover. In this article, we prove that if a semi-equivelar map is $k$ orbital then it has a finite index $m$-orbital minimal cover for $m \le k$. We also show the existence and classification of $n$-sheeted covers of semi-equivelar toroidal maps for each $n \in \mathbb{N}$

On the enumeration of a class of toroidal graphs

We present enumerations of a class of toroidal graphs which are called semi-equivelar maps. Semi-equivelar maps are generalizations of equivelar maps. There are eight non-isomorphic types of semi-equivelar maps on the torus: $\{3^{3}, 4^{2}\}$, $\{3^{2}, 4, 3, 4\}$, $\{3, 6, 3, 6\}$, $\{3^{4}, 6\}$, $\{4, 8^{2}\}$, $\{3, 12^{2}\}$, $\{4, 6, 12\}$, $\{3, 4, 6, 4\}$. We attempt to classify all these maps

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