4 research outputs found

    Uniform Star-factors of Graphs with Girth Three

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    A {\it star-factor} of a graph GG is a spanning subgraph of GG such that each component of which is a star. Recently, Hartnell and Rall studied a family U\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 U\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

    Star-uniform Graphs

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    A {\it star-factor} of a graph GG is a spanning subgraph of GG such that each of its component is a star. Clearly, every graph without isolated vertices has a star factor. A graph GG is called {\it star-uniform} if all star-factors of GG 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

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    A {\it star-factor} of a graph GG is a spanning subgraph of GG such that each component of which is a star. An {\it edge-weighting} of GG is a function w:E(G)⟶N+w: E(G)\longrightarrow \mathbb{N}^+, where N+\mathbb{N}^+ is the set of positive integers. Let Ω\Omega be the family of all graphs GG such that every star-factor of GG has the same weights under a fixed edge-weighting ww. In this paper, we present a simple structural characterization of the graphs in Ω\Omega that have girth at least five

    Uniform Star-factors of Graphs with Girth Three

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    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
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