Graphs in which all maximal bipartite subgraphs have the same order

Aequationes mathematicae volume 94, pages1241–1255(2020)Cite this article

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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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

Authors
  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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Keywords

  • Well-covered graph
  • Bipartite subgraph
  • Maximal

Mathematics Subject Classification

  • 05C69