Cycles in weighted graphs

Abstract

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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Bondy, J.A., Fan, G. Cycles in weighted graphs. Combinatorica 11, 191–205 (1991). https://doi.org/10.1007/BF01205072

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AMS subject classification (1980)

  • 05 C 35
  • 05 C 38