A note on quadrangular embedding of Abelian Cayley Graphs

A note on quadrangular embedding of Abelian Cayley Graphs$ J. E. Strapassonb,∗, S. I. R. Costaa, M. Muniz1 aInstitute of Mathematics, University of Campinas, 13083-970, Campinas, S˜ao Paulo, Brazil. bSchool of Applied Sciences, University of Campinas, 13484-350, Limeira, S˜ao Paulo, Brazil. cMathematics Department, Federal University of Paran´a, Curitiba, Paran´a, Brazil Abstract The genus graphs have been studied by many authors, but just a few results concerning in special cases: Planar, Toroidal, Complete, Bipartite and Cartesian Product of Bipartite. We present here a derive general lower bound for the genus of a abelian Cayley graph and construct a family of circulant graphs which reach this bound. Keywords: Abelian Cayley Graphs, Genus of a graph, Flat torus, Tessellations. 1. Introduction The genus of a graph, defined as the minimum genus of a 2-dimensional surface on which this graph can be embedded without crossings ([1, 2]), is well known as being an important measure of the graph complexity and it is related to other invariants. A circulant graph, Cn(a1, . . . , ak), is an homogeneous graph which can be represented (with crossings) by n vertices on a circle, with two vertices being connected if only if there is jump of ai vertices from one to the other (Figure 1). A circulant graph is particulary case of abelian Cayley graph. Different aspects of circulant graphs have been studied lately, either theoretically or through their applications in telecommunication networks and distributed computation [3, 4, 5, 6, 7, 8]. Concerning specifically to the genus of circulant graphs few results are known up to now. We quote [10] for a small class of toroidal (genus one) circulant graphs, [8] which establish a complete classification of planar circulant graphs, [9] which establish a complete classification of toroidal circulant graphs, and the cases where the circulant graph is either complete or a bipartite complete graph ([11, 12, 13, 14, 16]). In [17] we show how any circulant graph can be viewed as a quotient of lattices and obtain as consequences that: i) for k = 2, any circulant graph must be either genus one or zero (planar graph) and ii) for k = 3, there are circulant graphs of arbitrarily high genus. We present here a derive a general lower bound for the genus of abelian Cayley graph Cn(a1, . . . , ak) as (k −2) n + 4 4 , (Proposition 1), and construct a family of abelian Cayley graphs which reach this $Partially supported by FAPESP (Grants 2007/56052-8, 2007/00514-3 and 2011/01096-6) and CNPq (Grants 09561/2009-4) ∗Corresponding Author Email addresses: joao.strapasson@fca.unicamp.br (J. E. Strapasson), sueli@ime.unicamp.br (S. I. R. Costa), marcelo@mat.ufpr.br (M. Muniz) Preprint submitted to Arxiv October 31, 2018 arXiv:1004.0244v2 [math.GN] 4 Jan 2016 bound (Corollary 4). This note is organized as follows. In Section 2 we introduce concepts and previous results concerning circulant graphs, abelian Cayley graphs and genus. In Section 3 we derive a lower bound for the genus of an n-circulant graphs of order 2 k (Proposition 1) and construct families of graphs reaching this bound for arbitrarily k (Corollary 4). 2. Notation and Previous Results In this section we recall concepts and results used in this paper concerning acirculant graphs and fix the notations. Let G = ({e = g1, . . . gn}, +) be a finite abelian group. Given a subset S = {a1, . . . , ak} of G, the associated Cayley graph (G, S) is an undirected graph whose vertices are the elements of G, and where two vertices gi and gj are connected if and only if gi −gj = ±al for some al ∈S. We remark that (G, S) is connected if and only if S generates G as a group, and that this graph is 2k-regular if ai +ai ̸= 0, ∀i = 1, 2, . . . k, and 2k −l-regular otherwise, where l is a number of ai such that ai + ai = 0. A circulant graph Cn(a1, . . . , ak) with n vertices v0, . . . vn−1 and jumps a1, . . . , ak, 0 < aj ⩽ ⌊n/2⌋, ai ̸= aj, is an undirected graph such that each vertex vj, 0 ⩽j ⩽n −1, is adjacent to all the vertices vj±ai mod n, for 1 ⩽i ⩽k. A circulant graph is homogeneous: any vertex has the same order (number of incident edges), with is 2 k except when aj = n 2 for some j, when the order is 2 k −1, a circulant graphs is particulary case of abelian Cayley graph (G = Zn, S = {a1, . . . , ak}). The n-cyclic graph and the complete graph of n vertices are examples of circulant graphs denoted by Cn(1) and Cn(1, . . . , ⌊n/2⌋), respectively. Figure 1 shows on the left the standard picture of the circulant graph C13(1, 6). 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12 Figure 1: The circulant graph C13(1, 6) represented in the standard form (left) and on a 2-dimensional flat torus (right). In what follows we write (a1, . . . , ak) = (˜a1, . . . , ˜ak) mod n to indicate that for each i, there is j such that ai = ±˜aj mod n. Two circulant graphs, Cn(a1, . . . , ak) and Cn(˜a1, . . . , ˜ak) are said to satisfy the ´Ad´am’s relation if there is r, w

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