On the $\lambda$-fold Spectra for Bipartite Subgraphs of ${}^2{K}_4$

Abstract

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 $n$.