Uniform Star-factors of Graphs with Girth Three

TL;DR

This paper characterizes all graphs with girth three where every star-factor has the same number of edges, extending previous work on graphs with larger girth.

Contribution

It completely determines the family of graphs with girth three that have uniform star-factors, filling a gap in the existing research.

Findings

Identified all graphs with girth three having uniform star-factors.

Extended the classification of graphs with uniform star-factors to girth three.

Built upon previous results for girth at least five.

Abstract

A {\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each component of which is a star. Recently, Hartnell and Rall studied a family $\mathscr{U}$ of graphs satisfying the property that every star-factor of a member graph has the same number of edges. They determined the family $\mathscr{U}$ when the girth is at least five. In this paper, we investigate the family of graphs with girth three and determine all members of this family.