A note on heterogeneous decompositions into spanning trees

Abstract

In answer to a question of Eggleton, we prove that the complete multigraph on 5 vertices with edge multiplicity 6, namely $K_{5}^{(6)}$, has a decomposition into 5 copies of the family of trees of order 5 and that $K_{7}^{(22)}$ has a decomposition into 7 copies of the family of trees of order 7. We prove something similar for $K_{2n+1}$ for $n \le 13$.