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Register a new userSemi-equivelar maps on the torus are Archimedean
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If the face-cycles at all the vertices in a map on a surface are of same type then the map is called semi-equivelar. There are eleven types of Archimedean tilings on the plane. All the Archimedean tilings are semi-equivelar maps. If a map $X$ on the torus is a quotient of an Archimedean tiling on the plane then the map $X$ is semi-equivelar. We show that each semi-equivelar map on the torus is a quotient of an Archimedean tiling on the plane. Vertex-transitive maps are semi-equivelar maps. We know that four types of semi-equivelar maps on the torus are always vertex-transitive and there are examples of other seven types of semi-equivelar maps which are not vertex-transitive. We show that the number of ${rm Aut}(Y)$-orbits of vertices for any semi-equivelar map $Y$ on the torus is at most six. In fact, the number of orbits is at most three except one type of semi-equivelar maps. Our bounds on the number of orbits are sharp.
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A vertex-transitive map $X$ is a map on a surface on which the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. In particular, there are semi-equivelar maps on the torus, on the Klein bottle and on the surfaces of Euler characteristics $-1$ $&$ $-2$ which are not vertex-transitive. It is known that the boundaries of Platonic solids, Archimedean solids, regular prisms and antiprisms are vertex-transitive maps on $mathbb{S}^2$. Here we show that there is exactly one semi-equivelar map on $mathbb{S}^2$ which is not vertex-transitive. More precisely, we show that a semi-equivelar map on $mathbb{S}^2$ is the boundary of a Platonic solid, an Archimedean solid, a regular prism, an antiprism or the pseudorhombicuboctahedron. As a consequence, we show that all the semi-equivelar maps on $mathbb{RP}^2$ are vertex-transitive. Moreover, every semi-equivelar map on $mathbb{S}^2$ can be geometrized, i.e., every semi-equivelar map on $mathbb{S}^2$ is isomorphic to a semi-regular tiling of $mathbb{S}^2$. In the course of the proof of our main result, we present a combinatorial characterization in terms of an inequality of all the types of semi-equivelar maps on $mathbb{S}^2$. Here, we present self-contained combinatorial proofs of all our results.
A map $X$ on a surface is called vertex-transitive if the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called semi-equivelar. In general, semi-equivelar maps on a surface form a bigger class than vertex-transitive maps. There are semi-equivelar toroidal maps which are not vertex-transitive. In this article, we show that semi-equivelar toroidal maps are quotients of vertex-transitive toroidal maps. More explicitly, we prove that each semi-equivelar toroidal map has a finite vertex-transitive cover. In 2019, Drach {em et al.} have shown that each vertex-transitive toroidal map has a minimal almost regular cover. Therefore, semi-equivelar toroidal maps are quotients of almost regular toroidal maps.
A vertex-transitive map $X$ is a map on a closed surface on which the automorphism group ${rm Aut}(X)$ acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. We show that there are eleven types of semi-equivelar maps on the torus. Three of these are equivelar maps. It is known that two of the three types of equivelar maps on the torus are always vertex-transitive. We show that this is true for the remaining one type of equivelar map and one other type of semi-equivelar maps, namely, if $X$ is a semi-equivelar map of type $[6^3]$ or $[3^3, 4^2]$ then $X$ is vertex-transitive. We also show, by presenting examples, that this result is not true for the remaining seven types of semi-equivelar maps. There are ten types of semi-equivelar maps on the Klein bottle. We present examples in each of the ten types which are not vertex-transitive.
A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, and uniqueness, of a semi-regular tiling with a given vertex-type, and pose some open questions.
Trotter and Erdos found conditions for when a directed $m times n$ grid graph on a torus is Hamiltonian. We consider the analogous graphs on a two-holed torus, and study their Hamiltonicity. We find an $mathcal{O}(n^4)$ algorithm to determine the Hamiltonicity of one of these graphs and an $mathcal{O}(log(n))$ algorithm to find the number of diagonals, which are sets of vertices that force the directions of edges in any Hamiltonian cycle. We also show that there is a periodicity pattern in the graphs Hamiltonicities if one of the sides of the grid is fixed; and we completely classify which graphs are Hamiltonian in the cases where $n=m$, $n=2$, the $m times n$ graph has $1$ diagonal, or the $frac{m}{2} times frac{n}{2}$ graph has $1$ diagonal.
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