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
This is a preview of subscription content, log in to check access.
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
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
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