Cycles of weight divisible by $k$
Ajit A. Diwan
TL;DR
The paper studies zero-sum cycles in weight assignments modulo $k$ on graphs. It proves a new upper bound $f(k) < k+2\Omega(k)$ for the largest order of a weighted complete directed graph without a directed cycle of weight divisible by $k$, using a reduction lemma and a near arithmetic progression structure in $\mathbb{Z}_k$ to force a zero cycle; the bound is within one of the prime-case optimal, asymptotically close for large $k$. It also extends the undirected case to abelian-group weights, showing that any undirected graph with minimum degree at least $2k-1$ contains a zero cycle for weights from a group of order $k$, and discusses conjectured improvements to the bound (potentially $k+1$) with extremal examples, as well as a broader question about minimum degree and connectivity in directed graphs. The work advances zero-sum Ramsey-type results, providing explicit extremal bounds and a framework for extending to more general weight structures and graph classes, with several open conjectures highlighted for future research.
Abstract
A weighted (directed) graph is a (directed) graph with integer weights assigned to its vertices and edges. The weight of a subgraph is the sum of weights of vertices and edges in the subgraph. The problem of determining the largest order $f(k)$ of a weighted complete directed graph that does not contain a directed cycle of weight divisible by $k$, for an integer $k \ge 2$, was raised by Alon and Krivelevich [J. Graph Theory 98 (2021) 623-629]. They showed that $f(k)$ is $O(k\log k)$ and $f(k) \le 2k-2$ if $k$ is prime. The best bounds known to us are $f(k) \le 2k-2$ for all $k$ and $f(k) < (3k-1)/2$ for prime $k$. It is also known that $f(k) \ge k$ and this is believed to be the correct value. We prove that $f(k) < k+2Ω(k)$, where $Ω(k)$ is the number of prime factors, not necessarily distinct, in the prime factorization of $k$. We also show that any weighted undirected graph of minimum degree $2k-1$ contains a cycle of weight divisible by $k$. This result is proved in the more general setting in which the weights are from a finite abelian group of order $k$, and the cycle has weight equal to the group identity. We conjecture that this holds for undirected graphs with minimum degree $k+1$.