Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, November 05, 2019, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Charlotte Knierim

Long Cycles, Heavy Cycles and Cycle Decompositions in Directed Graphs

In this talk I show a connection between heavy cycles and cycle decompositions. We prove that every directed Eulerian graph can be decomposed into at most O(n log Δ) disjoint cycles, thus making progress towards the conjecture by Bollobás and Scott, and matching the best known upper bound from the undirected case. This also implies the existence of long cycles differing to the Erdős_Gallai bound for undirected graphs in only a log factor 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.

This is joint work with Maxime Larcher, Anders Martinsson and Andreas Noever.

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