Star-uniform Graphs

TL;DR

This paper characterizes star-uniform graphs, where all star-factors have the same number of components, using factor-criticality and domination number, addressing an open problem in graph theory.

Contribution

It provides a complete characterization of star-uniform graphs with minimum degree at least two, solving a problem posed by Hartnell and Rall.

Findings

Characterization of star-uniform graphs with minimum degree ≥ 2

Use of Gallai-Edmonds Matching Structure Theorem in the proof

Connection to minimum cost spanning trees and optimal assignment problems

Abstract

A {\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each of its component is a star. Clearly, every graph without isolated vertices has a star factor. A graph $G$ is called {\it star-uniform} if all star-factors of $G$ have the same number of components. To characterize star-uniform graphs was an open problem posed by Hartnell and Rall, which is motivated by the minimum cost spanning tree and the optimal assignment problems. We use the concepts of factor-criticality and domination number to characterize all star-uniform graphs with the minimum degree at least two. Our proof is heavily relied on Gallai-Edmonds Matching Structure Theorem.