  The small cycle decompositions of line graph ($\lambda$-fold line graph) of
complete graphs and complete bipartite graphs are studied by many papers. In
particular, Colby and Rodger obtained necessary and sufficient conditions for
the existence of a $C_{4}$-decomposition of the $\lambda$-fold line graph of
$K_{n}$, and Ganesamurthy and Paulraja completely determined the values of $n$
and $\lambda$ for which the $\lambda$-fold line graph of $K_{n}$ has a
$C_{5}$-decomposition. In this paper, we obtain the necessary and sufficient
condition for the existence of a $C_{4}$-decomposition of the $\lambda$-fold
line graph of $K_{m,n}$.
