A note on quadrangular embedding of Abelian Cayley Graphs

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.

💡 Research Summary

The paper investigates the genus of Abelian Cayley graphs when they are embedded so that every face of the embedding is bounded by at least four edges – a “quadrangular embedding”. After reviewing the classical definition of graph genus and summarizing prior work that largely focuses on special families (planar, toroidal, complete, bipartite, and Cartesian products of bipartite graphs), the authors set out to obtain a general lower bound for the genus of any Abelian Cayley graph.

Let (G = \mathbb{Z}{n_1}\times\cdots\times\mathbb{Z}{n_k}) be a finite Abelian group and let (S\subseteq G) be a symmetric generating set (i.e., (s\in S) implies (-s\in S)). The undirected Cayley graph (\mathrm{Cay}(G,S)) has (|G| = n) vertices and degree (|S| = d). In a quadrangular embedding each face must contain at least four edge‑incidences, which translates into the combinatorial identity (2E = 4F) (each edge belongs to two faces, each face contributes four edge‑incidences). Combining this with Euler’s formula (V - E + F = 2 - 2g) yields the inequality

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