Note on the connectivity of line graphs

  • Authors:

  • Angelika Hellwig;Dieter Rautenbach;Lutz Volkmann

  • Affiliations:

  • Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany;Forschungsinstitut für Diskrete Mathematik, University Bonn, 53113 Bonn, Germany;Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany

  • Venue:
  • Information Processing Letters - Devoted to the rapid publication of short contributions to information processing
  • Year:
  • 2004

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Abstract

Let G be a connected graph with vertex set V(G), edge set E(G), vertex-connectivity κ(G) and edge-connectivity λ(G). A subset S of E(G) is called a restricted edge-cut if G - S is disconnected and each component contains at least two vertices. The restricted edge-connectivity λ2(G) is the minimum cardinality over all restricted edge-cuts. Clearly λ2(G) ≥ λ(G) ≥ κ(G). In 1969, Chartrand and Stewart have shown that κ(L(G)) ≥ λ(G), if λ(G) ≥ 2. where L(G) denotes the line graph of G. In the present paper we show that κ(L(G)) = λ2(G), if |V (G)| ≥ 4 and G is not a star, which improves the result of Chartrand and Stewart. As a direct consequence of this identity, we obtain the known inequality λ2(G) ≤ ξ(G) by Esfahanian and Hakimi, where ξ(G) is the minimum edge degree.