A $C_4$-decomposition of the $\lambda$-fold line graph of $K_{m,n}$

Abstract

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 $K_{m,n}$.

13pages