2019/11/18 by Knierim, Charlotte, Larcher, Maxime, Martinsson, Anders +1 · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1911.07778
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), Dean (1986), and Bollobás and Scott (1996) 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 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.