Factorization graphs of finite groups

TL;DR

This paper introduces the concept of factorization graphs for finite groups, analyzing their structure, connectedness, and specific forbidden subgraph configurations, leading to classifications of groups based on these properties.

Contribution

It defines factorization graphs for finite groups and provides classifications for groups with connected, bipartite, claw-free, and square-free factorization graphs.

Findings

Characterized all finite groups with connected factorization graphs.

Classified groups with connected bipartite factorization graphs.

Identified groups with claw-free and square-free factorization graphs, excluding non-solvable groups.

Abstract

We introduce the factorization graph of a finite group and study its connectedness and forbidden structures. We characterize all finite groups with connected factorization graphs and classify those with connected bipartite factorization graphs. Also, we obtain a classification of all groups with claw-free, $K_{1,4}$-free, and square-free factorization graphs, excluding the non-solvable groups in the latter case.