On the \(\lambda\)-fold Spectra for Bipartite Subgraphs of \(^{2}K_{4}\)

For a graph \(H\) and a positive integer \(\lambda\), let \( ^{\lambda}{H} \) denote the multigraph obtained by replacing each edge of \(H\) with \(\lambda\) parallel edges. Let \(G\) be a multigraph with edge multiplicity \(2\) and with \(C_4\) as its underlying simple graph. We find necessary and sufficient conditions for the existence of a \(G\)-decomposition of \( ^{\lambda}{K_n} \) for all positive integers \(\lambda\) and \(n\).