Graphs in which all maximal bipartite subgraphs have the same order

  • Wayne GoddardEmail author
  • Kirsti Kuenzel
  • Eileen Melville


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.


Well-covered graph Bipartite subgraph Maximal 

Mathematics Subject Classification




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© 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

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