Forbidden subgraphs on conjugacy class graphs of groups

TL;DR

This paper studies the structure of conjugacy class graphs in finite groups, classifying forbidden subgraphs for various graph types across different classes of groups, and characterizing these graphs for specific well-known groups.

Contribution

It provides a comprehensive classification of forbidden induced subgraphs in conjugacy class graphs for multiple group classes, including symmetric, alternating, and simple groups.

Findings

Complete classifications for EPPO, order pq, and nilpotent groups.

Characterizations for symmetric, alternating, and certain solvable groups.

Solvable conjugacy class graphs are always cographs for specific groups.

Abstract

Let $G$ be a finite group. The \textit{commuting/nilpotent/solvable conjugacy class graph} ($\Gamma_{CCC}(G)$, $\Gamma_{NCC}(G)$, or $\Gamma_{SCC}(G)$) is a simple graph whose vertex set consists of all non-central conjugacy classes of $G$. Two vertices $x^G$ and $y^G$ are adjacent if and only if there exist elements $a \in x^G$ and $b \in y^G$ such that $\langle a, b \rangle$ forms an abelian, nilpotent, or solvable subgroup of $G$, respectively.\par In this paper, we mainly investigate cographs (it is $P_4$-free), chordal graphs (it is $C_n$-free $\forall \ n\ge 4$ ), split graphs (it contains no induced subgraph isomorphic to $C_4,\ C_5$, and $2K_2$), threshold graphs (it contains no induced subgraph isomorphic to $P_4$, $C_4,\ C_5$, and $2K_2$), and claw-free graphs (it contains no vertex with three pairwise non-adjacent neighbours) in terms of forbidden induced subgraphs in $\Gamma_{CCC}(G)$/ $\Gamma_{NCC}(G)$/$\Gamma_{SCC}(G)$.\par We provide a complete classification of these properties for EPPO groups, groups of order $pq$, and nilpotent groups. Additionally, we characterize the induced subgraphs in the commuting conjugacy class graph for symmetric and alternating groups. For solvable groups such as dihedral, dicyclic, and generalized dihedral groups, we establish complete results. Moreover, we fully characterize the graphs for the Mathieu groups $M_{11}$, $M_{12}$, and $M_{22}$, as well as certain minimal simple groups such as Suzuki groups and $\mathrm{PSL}(3,3)$. For other minimal simple groups, such as $\mathrm{PSL}(2,2^p)$, $\mathrm{PSL}(2,3^p)$, and $\mathrm{PSL}(2,p)$ (where $p > 3$ and $5 \mid p^2 + 1$), we demonstrate that the solvable conjugacy class graph is always a cograph. Finally, we present several open problems, highlighting further directions for research in this area.