Sufficient conditions for λ′-optimality in graphs with girth g

  • Authors:

  • C. Balbuena;P. García-Vázquez;X. Marcote

  • Affiliations:

  • Departament de Matemàtica Aplicada III Universitat Politècnica de Catalunya, Barcelona, Spain;Departamento de Matemática Aplicada I Universidad de Sevilla, Sevilla, Spain;Departament de Matemàtica Aplicada III Universitat Politècnica de Catalunya, Barcelona, Spain

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2006

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Abstract

For a connected graph the restricted edge-connectivityλ2(G) is defined as the minimum cardinality of anedge-cut over all edge-cuts S such that there are noisolated vertices in GS. A graph G is said tobe λ2-optimal if λ2(G) = ξ(G), whereξ(G) is the minimum edge-degree in G defined asξ(G) = min{d(u) + d(v)- 2:uv ε E(G)}, d(u)denoting the degree of a vertex u. A. Hellwig and L.Volkmann [Sufficient conditions for λ2-optimality in graphsof diameter 2, Discrete Math 283 (2004), 113120] gave a sufficientcondition for λ2-optimality in graphs of diameter 2. In thispaper, we generalize this condition in graphs of diameter g- 1, g being the girth of the graph, and show that a graphG with diameter at most g - 2 is λ2-optimal.© 2006 Wiley Periodicals, Inc. J Graph Theory 52: 7386,2006