Long cycles, heavy cycles and cycle decompositions in digraphs

Abstract

Hajós conjectured in 1968 that every Eulerian n-vertex graph can be decomposed into at most ⌊(n−1)/2⌋ edge-disjoint cycles. This has been confirmed for some special graph classes, but the general case remains open. In a sequence of papers by Bienia and Meyniel (1986) [1], Dean (1986) [7], and Bollobás and Scott (1996) [2] it was analogously conjectured that every directed Eulerian graph can be decomposed into O(n) cycles.In this paper, we show that every directed Eulerian graph can be decomposed into O(nlog⁡Δ) disjoint cycles, thus making progress towards the conjecture by Bollobás and Scott. Our approach is based on finding heavy cycles in certain edge-weightings of directed graphs. As a further consequence of our techniques, we prove that for every edge-weighted digraph in which every vertex has out-weight at least 1, there exists a cycle with weight at least Ω(log⁡log⁡n/log⁡n), thus resolving a question by Bollobás and Scott.