4 research outputs found
Uniform Star-factors of Graphs with Girth Three
A {\it star-factor} of a graph is a spanning subgraph of such that
each component of which is a star. Recently, Hartnell and Rall studied a family
of graphs satisfying the property that every star-factor of a
member graph has the same number of edges. They determined the family
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
Star-uniform Graphs
A {\it star-factor} of a graph is a spanning subgraph of such that
each of its component is a star. Clearly, every graph without isolated vertices
has a star factor. A graph is called {\it star-uniform} if all star-factors
of 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
Uniformly Weighted Star-Factors of Graphs
A {\it star-factor} of a graph is a spanning subgraph of such that
each component of which is a star. An {\it edge-weighting} of is a function
, where is the set of
positive integers. Let be the family of all graphs such that every
star-factor of has the same weights under a fixed edge-weighting . In
this paper, we present a simple structural characterization of the graphs in
that have girth at least five
Uniform Star-factors of Graphs with Girth Three
A 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 U of graphs satisfying the property that every star-factor of a member graph has the same number of edges. They determined the family 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. Key words: star-factor, uniform star-factor, girth, edge-weighting