  Let $G$ be a $q$-regular bipartite graph with bipartition $(U,V)$. It was
proved by Lu, Wang, and Yan in 2020 that $G$ has a spanning subgraph $H$ such
that each vertex of $U$ has degree 1 in $H$, and each vertex of $V$ has degree
distinct from 1 in $H$. We extend the result to multigraphs, under the
condition that $q$ is a prime power and the number of perfect matchings of $G$
is not divisible by $q$. The condition on the number of perfect matchings is
necessary for multigraphs.
  We conclude with a conjecture on the limiting distribution of the number of
perfect matchings modulo $q$ in a random bipartite $q$-regular graph.
