Notes on heavy cycles in weighted digraphs

A weighted digraph is a digraph such that every arc is assigned a nonnegative number, called the weight of the arc. The weighted outdegree of a vertex v in a weighted digraph D is the sum of the weights of the arcs with v as their tail, and the weight of a directed cycle C in D is the sum of the weights of the arcs of C . In this note we prove that if every vertex of a weighted digraph D with order n has weighted outdegree at least 1, then there exists a directed cycle in D with weight at least 1 / log 2 n . This proves a conjecture of Bollobلs and Scott up to a constant factor.