Uniformly Weighted Star-Factors of Graphs

TL;DR

This paper characterizes graphs with girth at least five where all star-factors have identical edge-weight sums, providing a structural understanding of such uniformly weighted star-factors.

Contribution

It offers a simple structural characterization of graphs with girth at least five where all star-factors share the same total edge weight.

Findings

Graphs in the family  have a specific structure.

All star-factors in these graphs have equal total weights.

The characterization applies to graphs with 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. An {\it edge-weighting} of $G$ is a function $w: E(G)\longrightarrow \mathbb{N}^+$, where $\mathbb{N}^+$ is the set of positive integers. Let $\Omega$ be the family of all graphs $G$ such that every star-factor of $G$ has the same weights under a fixed edge-weighting $w$. In this paper, we present a simple structural characterization of the graphs in $\Omega$ that have girth at least five.