A $C_{4}$-decomposition of the $λ$-fold line graph of $K_{m,n}$

The small cycle decompositions of line graph ($λ$-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 $λ$-fold line graph of $K_{n}$, and Ganesamurthy and Paulraja completely determined the values of $n$ and $λ$ for which the $λ$-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 $λ$-fold line graph of $K_{m,n}$.

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