Genus of commuting conjugacy class graph of groups

Abstract

For a non-abelian group $G$, its commuting conjugacy class graph $\mathcal{CCC}(G)$ is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$ are adjacent if there exists some elements $x^{\prime }\in x^G$ and $y^{\prime }\in y^G$ such that $x^{\prime }{y}^{\prime }=y^{\prime }{x}^{\prime }$. In this paper we compute the genus of $\mathcal{CCC}(G)$ for six well-known classes of non-abelian two-generated groups (viz. $D_{2n},S{D}_{8n},Q_{4m},V_{8n},U_{(n,m)}$ and $G(p,m,n)$) and determine whether $\mathcal{CCC}(G)$ for these groups are planar, toroidal, double-toroidal or triple-toroidal.

12 pages