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Star-Uniform Graphs

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Abstract

A star-factor of a graph is a spanning subgraph each of whose components is a star. A graph G is called star-uniform if all star-factors of G have the same number of components. Motivated by the minimum cost spanning tree and the optimal assignment problems, Hartnell and Rall posed an open problem to characterize all the star-uniform graphs. In this paper, we show that a graph G is star-uniform if and only if G has equal domination and matching number. From this point of view, the star-uniform graphs were characterized by Randerath and Volkmann. Unfortunately, their characterization is incomplete. By deploying Gallai–Edmonds Matching Structure Theorem, we give a clear and complete characterization of star-unform graphs.

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Authors and Affiliations

  1. Department of Computer and Information Sciences, Ibaraki University, Hitachi, Ibaraki, Japan

    Mikio Kano

  2. Department of Mathematics, Southeast University, Nanjing, 211189, China

    Yunjian Wu

  3. Department of Mathematics and Statistics, Thompson Rivers University, Kamloops, BC, Canada

    Qinglin Yu

Authors
  1. Mikio Kano
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  2. Yunjian Wu
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  3. Qinglin Yu
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Corresponding author

Correspondence to Yunjian Wu.

Additional information

Qinglin Yu was supported in part by 973 Project of Ministry of Science and Technology of China and Natural Sciences and Engineering Research Council of Canada.

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Kano, M., Wu, Y. & Yu, Q. Star-Uniform Graphs. Graphs and Combinatorics 26, 383–394 (2010). https://doi.org/10.1007/s00373-010-0917-x

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  • DOI: https://doi.org/10.1007/s00373-010-0917-x

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