On Heterogeneous Decompositions of Uniform Complete Multigraphs into Spanning Trees

Abstract

Let be the order n uniform complete multigraph with edge multiplicity r. A spanning tree decomposition of partitions its edge set into a family J of edge-induced spanning trees. In a purely heterogeneous decomposition J no trees are isomorphic. Every order n tree occurs in a fully heterogeneous decomposition J. All trees have equal multiplicity in a balanced decomposition J. We show: (1) has 16 decomposition classes, four of which contain the 24 fully heterogeneous decompositions; (2) has 34 fully heterogeneous decomposition classes; exactly one lacks decompositions reducible to two decompositions; (3) when k ≥ 1, many balanced fully heterogeneous decompositions of reduce to “smaller” decompositions, but one such decomposition of is irreducible.