$K_{3,3}$-free Intersection Graphs of Finite Groups

TL;DR

This paper classifies all finite groups whose intersection graphs of proper non-trivial subgroups do not contain a complete bipartite subgraph K_{3,3}.

Contribution

It provides a complete classification of finite groups with K_{3,3}-free intersection graphs, a new structural insight into subgroup intersections.

Findings

Identifies all finite groups with K_{3,3}-free intersection graphs.

Characterizes the subgroup intersection patterns avoiding K_{3,3}.

Contributes to understanding the structure of subgroup intersection graphs.

Abstract

The intersection graph of a group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$ and $K$ if and only if $H\cap K \neq 1$ where $1$ denotes the trivial subgroup of $G$. In this paper we classify all finite groups whose intersection graphs are $K_{3,3}$-free.