Advertisement

Graphs in which all maximal bipartite subgraphs have the same order

  • Wayne GoddardEmail author
  • Kirsti Kuenzel
  • Eileen Melville
Article
  • 11 Downloads

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.

Keywords

Well-covered graph Bipartite subgraph Maximal 

Mathematics Subject Classification

05C69 

Notes

References

  1. 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)MathSciNetzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hartnell, B.L., Rall, D.F.: On the Cartesian product of non well-covered graphs. Electron. J. Comb. 20(2) (2013) Paper 21, 4Google Scholar
  4. 4.
    Plummer, M.D.: Some covering concepts in graphs. J. Comb. Theory 8, 91–98 (1970)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Prisner, E., Topp, J., Vestergaard, P.D.: Well covered simplicial, chordal, and circular arc graphs. J. Graph Theory 21, 113–119 (1996)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ravindra, G.: Well-covered graphs. J. Comb. Inf. Syst. Sci. 2, 20–21 (1977)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Topp, J., Volkmann, L.: On the well-coveredness of products of graphs. Ars Comb. 33, 199–215 (1992)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Zhu, X.: Bipartite subgraphs of triangle-free subcubic graphs. J. Comb. Theory Ser. B 99, 62–83 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Wayne Goddard
    • 1
    Email author
  • Kirsti Kuenzel
    • 2
  • Eileen Melville
    • 1
  1. 1.School of Mathematical and Statistical SciencesClemson UniversityClemsonUSA
  2. 2.Department of MathematicsTrinity CollegeHartfordUSA

Personalised recommendations