Cycles in weighted graphs


A weighted graph is one in which each edgee is assigned a nonnegative numberw(e), called the weight ofe. The weightw(G) of a weighted graphG is the sum of the weights of its edges. In this paper, we prove, as conjectured in [2], that every 2-edge-connected weighted graph onn vertices contains a cycle of weight at least 2w(G)/(n−1). Furthermore, we completely characterize the 2-edge-connected weighted graphs onn vertices that contain no cycle of weight more than 2w(G)/(n−1). This generalizes, to weighted graphs, a classical result of Erdős and Gallai [4].

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  1. [1]

    J. A. Bondy: Trigraphs,Discrete Math. 75 (1989), 69–79.

    Google Scholar 

  2. [2]

    J. A. Bondy, andG. Fan: Optimal paths and cycles in weighted graphs, Graph Theory in Memory of G. A. Dirac,Annals of Discrete Mathematics,41 (1989), 53–70.

    Google Scholar 

  3. [3]

    J. A. Bondy, andS. C. Locke: Relative lengths of paths and cycles in 3-connected graphs,Discrete Math. 33 (1981), 111–122.

    Google Scholar 

  4. [4]

    P. Erdős, andT. Gallai: On maximal paths and circuits of graphs,Acta Math. Acad. Sci. Hungar. 10 (1959), 337–356.

    Google Scholar 

  5. [5]

    A. Frieze, C. McDiarmid, andB. Reed: On a conjecture of Bondy and Fan, manuscript.

  6. [6]

    P. D. Seymour: Sums of circuits,Graph Theory and Related Topics (edited by J. A. Bondy and U. S. R. Murty), Academic Press, New York, San Francisco, London (1979), 341–355.

    Google Scholar 

  7. [7]

    P. D. Seymour: Personal communication.

  8. [8]

    G. Szekeres: Polyhedral decompositions of cubic graphs,Bull. Austral. Math. Soc. 8 (1973), 367–387.

    Google Scholar 

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Bondy, J.A., Fan, G. Cycles in weighted graphs. Combinatorica 11, 191–205 (1991).

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AMS subject classification (1980)

  • 05 C 35
  • 05 C 38