Lower bounds on the vertex-connectivity of digraphs and graphs

  • Authors:

  • Angelika Hellwig;Lutz Volkmann

  • Affiliations:

  • Institute of Medical Statistics, Medical Faculty RWTH Aachen, Aachen, Germany;Lehrstuhl II für Mathematik, RWTH Aachen University, Aachen, Germany

  • Venue:
  • Information Processing Letters
  • Year:
  • 2006

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Abstract

Since interconnection networks are often modeled by graphs or digraphs, the connectivity of a (di-)graph is an important measurement for fault tolerance of networks.Let G be a graph of order n, minimum degree δ, and vertex-connectivity k. If G is not the complete graph, then Chartrand and Harary [G. Chartrand, F. Harary, Graphs with prescribed connectivities, in: P. Erdös, G. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp. 61-63] proved in 1968 that k≥2δ+2-n.In 1993, Topp and Volkmann [J. Topp, L. Volkmann, Sufficient conditions for equality of connectivity and minimum degree of a graph, J. Graph Theory 17 (1993) 695-700] proved the following analog bound for bipartite graphs. If G is a bipartite graph with kk≥4δ-n.In this paper we present some generalizations and extensions of these inequalities for digraphs as well as for graphs.