Hajós conjectured in 1968 that every Eulerian n-vertex graph can be decomposed into at most $\lfloor (n-1)/2\rfloor$ 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 $O(n\log \mathrm{\Delta })$ 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 $\mathrm{\Omega }(\log \log n/\log n)$, thus resolving a question by Bollobás and Scott.

Hajós在1968年推测，每个欧拉n-顶点图最多可以分解为$\lfloor （ñ-\mathrm{1个}）/2\rfloor$边不相交的循环。对于某些特殊的图类，已经确认了这一点，但是一般情况仍然存在。在Bienia和Meyniel（1986）[1]，Dean（1986）[7]，Bollobás和Scott（1996）[2]的一系列论文中，类似地推测每个有向欧拉图都可以分解为$Ø（ñ）$ 周期。

在本文中，我们证明了每个有向欧拉图都可以分解为 $Ø（ñ\mathrm{日志}\mathrm{\Delta }）$不相交的周期，因此在Bollobás和Scott的猜想中取得了进展。我们的方法是基于在有向图的某些边加权中找到重周期。我们的技术的进一步结果是，我们证明对于每个顶点权重至少为1的边加权图，存在一个权重至少为1的循环$\mathrm{\Omega }（\mathrm{日志}\mathrm{日志}ñ/\mathrm{日志}ñ）$，因此解决了Bollobás和Scott的问题。