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Register a new userVertex-transitive covers of semi-equivelar toroidal maps
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Authors
Basudeb Datta
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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.
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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.
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.
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.
We introduce the notion of a symmetric basis of a vector space equipped with a quadratic form, and provide a sufficient and necessary condition for the existence to such a basis. Symmetric bases are then used to study Cayley graphs of certain extraspecial 2-groups of order 2^{2r+1} (rgeq 1), which are further shown to be normal Cayley graphs and 2-arc-transitive covers of 2r-dimensional hypercubes.
We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex transitive graph. As an intermediate step, we prove that every countably infinite, connected, vertex transitive graph has a perfect matching. Incidentally, we construct an example of a 2-ended cubic vertex transitive graph which is not a Cayley graph, answering a question of Watkins from 1990.
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