  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 $f(k) < k+2\Omega(k)$, where $\Omega(k)$ 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 $k+1$.
