Forbidden subgraphs on conjugacy class graphs of groups

TL;DR

The paper addresses the problem of characterizing forbidden induced subgraphs in three conjugacy-class graphs $\Gamma_{CCC}(G)$, $\Gamma_{NCC}(G)$, and $\Gamma_{SCC}(G)$ for finite groups, where adjacency is defined by abelian, nilpotent, or solvable generated subgroups. It develops a comprehensive program to classify these graphs across broad group families, including EPPO groups, groups of order $pq$, nilpotent groups, symmetric/alternating groups, several solvable groups (like dihedral and dicyclic), and various simple groups (Suzuki, Mathieu, and finite minimal simple groups), with further analysis via invariably generated subgroups for PSL-type minimal simple groups. The main contributions are complete or near-complete forbidden-subgraph characterizations (e.g., $P_4$-free cographs, $C_n$-free chordal, $2K_2$-free, and claw-free graphs) for many families, along with explicit structural descriptions and, in several cases, computational verifications (notably for Mathieu groups). These results illuminate how group-theoretic structure dictates global graph properties, enabling rigorous classification of graph families arising from conjugacy classes and suggesting directions for future work in minimal simple groups and broader solvable groups. The work thus advances the interface between finite group theory and graph-theoretic forbidden-subgraph theory with potential applications in algebraic combinatorics and related algorithmic classifications.

Abstract

Let $G$ be a finite group. The \textit{commuting/nilpotent/solvable conjugacy class graph} ($Γ_{CCC}(G)$, $Γ_{NCC}(G)$, or $Γ_{SCC}(G)$) 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 $Γ_{CCC}(G)$/ $Γ_{NCC}(G)$/$Γ_{SCC}(G)$.\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.

Theorems & Definitions (45)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof