Isomorphic Factorization

A graph H divides a graph G if the edge set of G can be partitioned into copies of H.

Factoring Complete Graphs into Trivalent Graphs

There are 19 trivalent graphs of order 10. Of these, 14 divide K10. The five that don't are shown in the figure below.

Factoring Complete Graphs into Spanning Trees

In order for a tree T of order n to divide the complete graph Kn, n must be even and the maximum degree of T can be at most n/2. The degree sequence of a tree of order n does not determine whether or not it divides Kn. There are two trees with degree sequence 3,2,2,1,1,1. They are shown in the figure below. The top tree does not divide K6 while the bottom one does.
The table below gives some data from computational experiments.
Order # Trees # Trees of Max Deg n/2 or less # Trees that Divide Kn
4 2 1 1
6 6 4 3
8 23 18 14
10 106 94 85
12 551 520 484+