A Fundamental Theorem of Multigraph Decomposition of a \(\lambda K_{m,n}\)

Much research has been done on the edge decomposition of \(\lambda\) copies of the complete graph \(G\) with respect to some specified subgraph \(H\) of \(G\). This is equivalent to the investigation of \((G, H)\)-designs of index \(\lambda\). In this paper, we present a fundamental theorem on the decomposition of \(\lambda\) copies of a complete bipartite graph. As an application of this result, we show that necessary conditions are sufficient for the decomposition of \(\lambda\) copies of a complete bipartite graph into several multi-subgraphs \(H\) with a number of vertices less than or equal to \(4\) and the number of edges less than or equal to \(4\), with some exceptions where decompositions do not exist. These decomposition problems are interesting to study as various decompositions do not exist even when necessary conditions are satisfied.