Long Cycles, Heavy Cycles and Cycle Decompositions in Digraphs

TL;DR

This paper advances the understanding of cycle decompositions in directed Eulerian graphs, showing they can be decomposed into O(n log Δ) cycles and identifying heavy cycles in weighted digraphs, addressing longstanding conjectures.

Contribution

It proves that directed Eulerian graphs can be decomposed into O(n log Δ) cycles and establishes the existence of heavy cycles in certain weighted digraphs, improving previous bounds.

Findings

Directed Eulerian graphs decompose into O(n log Δ) cycles.

Existence of cycles with weight at least Ω(log log n / log n) in weighted digraphs.

Progress towards longstanding conjectures on cycle decompositions.

Abstract

Haj\'os 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), Dean (1986), and Bollob\'as and Scott (1996) it was analogously conjectured that every \emph{directed} Eulerian graph can be decomposed into $O(n)$ cycles. In this paper, we show that every directed Eulerian graph can be decomposed into $O(n \log \Delta)$ disjoint cycles, thus making progress towards the conjecture by Bollob\'as 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 $\Omega(\log \log n/{\log n})$, thus resolving a question by Bollob\'as and Scott.