Cycles of weight divisible by k

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 ≥ 2, was raised by Alon and Krivelevich [J. Graph Theory 98 (2021) 623-629]. They showed that $f(k)$ is O(klog k) and f(k) ≤ 2k-2 if k is prime. The best bounds known to us are f(k) ≤ 2k-2 for all k and $f(k) < (3k-1)/2$ for prime k. It is also known that f(k) ≥ 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 $k+1$.