Uniformly Weighted Star-Factors of Graphs

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 $w:E(G)⟶{\mathbb{N}}^{+}$, where ${\mathbb{N}}^{+}$ is the set of positive integers. Let $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ that have girth at least five.