Zero sum cycles in complete digraphs

Abstract

Given a non-trivial finite Abelian group (A,+), let n(A)≥2 be the smallest integer such that for every labelling of the arcs of the bidirected complete graph K↔n(A) with elements from A there exists a directed cycle for which the sum of the arc-labels is zero. The problem of determining n(Zq) for integers q≥2 was recently considered by Alon and Krivelevich, (2020), who proved that n(Zq)=O(qlogq). Here we improve their result and show that n(Zq) grows linearly. More generally we prove that for every finite Abelian group A we have n(A)≤8|A|, while if |A| is prime then n(A)≤32|A|.As a corollary we obtain that every K16q-minor contains a cycle of length divisible by q for every integer q≥2, which improves a result from Alon and Krivelevich, (2020).