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Graphs in which all maximal bipartite subgraphs have the same order

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Abstract

Motivated by the concept of well-covered graphs, we define a graph to be well-bicovered if every vertex-maximal bipartite subgraph has the same order (which we call the bipartite number). We first give examples of them, compare them with well-covered graphs, and characterize those with small or large bipartite number. We then consider graph operations including the union, join, and lexicographic and cartesian products. Thereafter we consider simplicial vertices and 3-colored graphs where every vertex is in triangle, and conclude by characterizing the maximal outerplanar graphs that are well-bicovered.

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References

  1. Campbell, S.R., Ellingham, M.N., Royle, G.F.: A characterisation of well-covered cubic graphs. J. Comb. Math. Comb. Comput. 13, 193–212 (1993)

    MathSciNet  MATH  Google Scholar 

  2. Finbow, A., Hartnell, B., Nowakowski, R.J.: A characterization of well covered graphs of girth \(5\) or greater. J. Comb. Theory Ser. B 57, 44–68 (1993)

    Article  MathSciNet  Google Scholar 

  3. Hartnell, B.L., Rall, D.F.: On the Cartesian product of non well-covered graphs. Electron. J. Comb. 20(2) (2013) Paper 21, 4

  4. Plummer, M.D.: Some covering concepts in graphs. J. Comb. Theory 8, 91–98 (1970)

    Article  MathSciNet  Google Scholar 

  5. Prisner, E., Topp, J., Vestergaard, P.D.: Well covered simplicial, chordal, and circular arc graphs. J. Graph Theory 21, 113–119 (1996)

    Article  MathSciNet  Google Scholar 

  6. Ravindra, G.: Well-covered graphs. J. Comb. Inf. Syst. Sci. 2, 20–21 (1977)

    MathSciNet  MATH  Google Scholar 

  7. Topp, J., Volkmann, L.: On the well-coveredness of products of graphs. Ars Comb. 33, 199–215 (1992)

    MathSciNet  MATH  Google Scholar 

  8. Zhu, X.: Bipartite subgraphs of triangle-free subcubic graphs. J. Comb. Theory Ser. B 99, 62–83 (2009)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, USA

    Wayne Goddard & Eileen Melville

  2. Department of Mathematics, Trinity College, Hartford, CT, USA

    Kirsti Kuenzel

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  1. Wayne Goddard
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  2. Kirsti Kuenzel
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  3. Eileen Melville
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Correspondence to Wayne Goddard.

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Goddard, W., Kuenzel, K. & Melville, E. Graphs in which all maximal bipartite subgraphs have the same order. Aequat. Math. 94, 1241–1255 (2020). https://doi.org/10.1007/s00010-020-00700-x

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  • DOI: https://doi.org/10.1007/s00010-020-00700-x

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