On the planarity of cyclic graphs (original) (raw)

2018, Rocky Mountain Journal of Mathematics

We classify all finite groups with planar cyclic graphs. Also, we compute the genus and crosscap number of some families of groups (by knowing that of the cyclic graph of particular proper subgroups in some cases). 1. Introduction. Let G be a group. For each x ∈ G, the cyclizer of x is defined as Cyc G (x) = {y ∈ G | ⟨x, y⟩ is cyclic}. In addition, the cyclizer of G is defined by Cyc(G) = ∩ x∈G Cyc G (x). Cyclizers were introduced by Patrick and Wepsic in [15] and studied in [1, 2, 3, 9, 14, 15]. It is known that Cyc(G) is always cyclic and that Cyc(G) ⊆ Z(G). In particular, Cyc(G) G. The cyclic graph (respectively, weak cyclic graph) of a group G is the simple graph with vertex-set G\Cyc(G) (respectively, G\{1}) such that two distinct vertices x and y are adjacent if and only if ⟨x, y⟩ is cyclic. The cyclic graph and weak cyclic graph of G are denoted by Γ c (G) and Γ w c (G), respectively. From the explanation above, Γ c (G) (respectively, Γ w c (G)) is the null graph if G is cyclic (respectively, trivial). Thus, we will assume that G is non-cyclic (respectively, nontrivial) when working with Γ c (G) (respectively, Γ w c (G)). A graph is planar if it can be drawn in the plane in such a way that its edges intersect only at the end vertices. Recall that a subdivision of an edge {u, v} in a graph Γ is the replacement of the edge {u, v} in Γ with two new edges {u, w} and {w, v} in which w is a new vertex.