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Zero sum cycles in complete digraphs
European Journal of Combinatorics ( IF 1.0 ) Pub Date : 2021-07-20 , DOI: 10.1016/j.ejc.2021.103399
Tamás Mészáros 1 , Raphael Steiner 2
Affiliation  

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 Kn(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 q2 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 q2, which improves a result from Alon and Krivelevich, (2020).



中文翻译:

完全有向图中的零和循环

给定一个非平凡的有限阿贝尔群 (一种,+), 让 n(一种)2 是最小整数,使得对于双向完全图的弧的每个标记 n(一种) 与元素来自 一种存在一个有向循环,其中弧标签的总和为零。确定的问题n(Zq) 对于整数 q2 最近被 Alon 和 Krivelevich (2020) 考虑,他们证明了 n(Zq)=(q日志q). 在这里,我们改进了他们的结果并表明n(Zq)线性增长。更一般地,我们证明对于每个有限阿贝尔群一种 我们有 n(一种)8|一种|, 而如果 |一种| 那么是素数 n(一种)32|一种|.

作为推论,我们得到每个 16q-minor 包含一个长度可被整除的循环 q 对于每个整数 q2,这改进了 Alon 和 Krivelevich (2020) 的结果。

更新日期:2021-07-20
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