Finite groups whose noncyclic graphs have positive genus
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FINITE GROUPS WHOSE NONCYCLIC GRAPHS HAVE POSITIVE GENUS X. MA1,† and H. SU2,∗,‡ 1
2
School of Science, Xi’an Shiyou University, Xi’an, Shaanxi, 710065, China e-mail: [email protected]
School of Mathematics and Statistics, Nanning Normal University, Nanning, Guangxi, 530299, China e-mail: [email protected] (Received January 1, 2020; revised January 9, 2020; accepted January 13, 2020)
Abstract. For a finite noncyclic group G, let Cyc(G) be a set of elements a of G such that �a, b� is cyclic for each b of G. The noncyclic graph of G is a graph with the vertex set G \ Cyc(G), having an edge between two distinct vertices x and y if �x, y� is not cyclic. In this paper, we show that, for a fixed nonnegative integer k, there are at most finitely many finite noncyclic groups whose noncyclic graphs have (non)orientable genus k. We also classify the finite noncyclic groups whose noncyclic graphs have (non)orientable genus 1, 2, 3 and 4, respectively.
1. Introduction and results Every graph considered in this paper is finite, undirected, with no loops and no multiple edges. Let Γ be a graph. The vertex set of Γ is denoted by V (Γ), and the edge set by E(Γ). An embedding of a graph into a surface is a drawing of the graph on the surface in such a way that its edges may intersect only at their endpoints. A graph is said to be planar if it can be embedded in the plane. A non-planar graph can be embedded in some surface obtained from the sphere by attaching some handles or crosscaps. We denote by Sk a sphere with k handles and by Nk a sphere with k crosscaps. Note that both S0 and N0 are the sphere itself, and S1 and N1 are a torus and a projective plane, respectively. The smallest non-negative integer k ∗ Corresponding
author. first author was supported by the National Natural Science Foundation of China (Grant No. 11801441), and the Young Talent fund of University Association for Science and Technology in Shaanxi, China (Program No. 20190507). ‡ The second author was supported by the National Natural Science Foundation of China (Grant No. 11661013). Key words and phrases: genus, noncyclic graph, finite group. Mathematics Subject Classification: 05C25, 05C10. † The
c 2020 0236-5294/$ 20.00 © 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
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X. X. MA MA and and H. H. SU SU
such that a graph Γ can be embedded in Sk is called the orientable genus or genus of Γ, and is denoted by γ(Γ). The nonorientable genus of Γ, denoted by γ(Γ), is the smallest integer k such that Γ can be embedded in Nk . The problem of finding the graph genus is NP-hard [41]. Many research articles have appeared on the genus of graphs constructed from some algebraic structures. For example, Wang [42] found two families of finite rings whose zero-divisor graphs have genus at most one. Chiang-Hsieh et al. [11] classified the finite commutative rings whose zero-divisor graphs have genus at most one. Bloomfield and Wickham [7] enumerated the local finite commutative rings whose zero divisor graphs hav
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