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

TL;DR

This paper establishes the precise conditions under which the $ ext{C}_4$ cycle can decompose the $ ext{lambda}$-fold line graph of complete bipartite graphs, extending previous work on complete and complete bipartite graphs.

Contribution

It provides the first necessary and sufficient conditions for a $C_4$-decomposition of the $ ext{lambda}$-fold line graph of $K_{m,n}$, filling a gap in graph decomposition theory.

Findings

Derived necessary and sufficient conditions for $C_4$-decomposition.

Extended previous results from complete graphs to bipartite graphs.

Contributed to the understanding of cycle decompositions in complex graph structures.

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 $K_{m,n}$.