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

Date and Time: Tuesday, May 07, 2024, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Kalina Petrova

A note on digraph splitting

A tantalizing open problem, posed independently by Stiebitz in 1995 and by Alon in 2006, asks whether for every pair of integers s, t ≥ 1 there exists a finite number F(s, t) such that the vertex set of every digraph of minimum out-degree at least F(s, t) can be partitioned into non-empty parts A and B such that the subdigraphs induced on A and B have minimum out-degree at least s and t, respectively. We prove that if F(2, 2) exists, then all the numbers F(s, t) with s, t ≥ 1 exist and satisfy F(s, t) = Θ(s + t). In consequence, the problem of Alon and Stiebitz reduces to the case s = t = 2. Moreover, the numbers F(s, t) with s, t ≥ 2 either all exist and grow linearly, or all of them do not exist. This is joint work with Micha Christoph and Raphael Steiner.

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