Resolvable tree designs (Q1200005)

A graph \(G\) has an \(H\)-factorization \((H| G)\) if the edges of \(G\) can be partitioned into \(H\)-factors (an \(H\)-factor is a spanning subgraph each component of which is isomorphic to \(H\)). Let \(K_ n\) denote the complete graph of order \(n\) where each edge has multiplicity \(\lambda\) and \(T_ k\) be a tree with \(k\) vertices. Main results are \[ \text{If } T_ k| K_ k\text{ then } T_ k|\lambda K_ n\text{ iff } n\equiv 0\pmod k\text{ and } \lambda k(n-1)\equiv 0\pmod{2(k-1)}. \tag{1} \] \[ \text{If } T_ k| 2K_ k\text{ then }T_ k| 2\lambda K_ n\text{ iff } n\equiv 0\pmod k \text{ and }\lambda k(n-1)\equiv 0\pmod{k-1}. \tag{2} \] {}.