# 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 K_{10}. 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 K_{n},
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 K_{n}.
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 K_{6} 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 K_{n} |

4 |
2 |
1 |
1 |

6 |
6 |
4 |
3 |

8 |
23 |
18 |
14 |

10 |
106 |
94 |
85 |

12 |
551 |
520 |
484+ |