A note on heavy cycles in weighted digraphs

TL;DR

This paper proves that in weighted digraphs with minimum outdegree 1, there exists a cycle with weight at least 1/log_2(n), confirming a conjecture up to a constant factor.

Contribution

It establishes a lower bound on the weight of cycles in weighted digraphs with minimum outdegree, confirming a conjecture by Bollobás and Scott up to a constant.

Findings

Existence of a cycle with weight at least 1/log_2(n)

Confirms Bollobás and Scott's conjecture up to a constant

Provides bounds on cycle weights in weighted digraphs

Abstract

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 Bollob\'{a}s and Scott up to a constant factor.