Long cycles, heavy cycles and cycle decompositions in digraphs

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 classes, but the general case remains open. In a sequence of papers by Bienia and Meyniel (1986) [1], Dean [7], Bollobás Scott (1996) [2] it was analogously directed O(n) this paper, we show O(nlog⁡Δ) disjoint cycles, thus making progress towards conjecture Scott. Our approach is based on finding heavy cycles certain edge-weightings graphs. As further consequence our techniques, prove edge-weighted digraph which vertex out-weight least 1, there exists cycle with weight Ω(log⁡log⁡n/log⁡n), resolving question