### edge-disjoint isomorphic multicolored trees and cycles in complete

```EDGE-DISJOINT ISOMORPHIC
MULTICOLORED TREES AND CYCLES
IN COMPLETE GRAPHS

9622053

Abstract



Author
Gregory M. Constantine
Publisher
Society for Industrial and Applied
Year of Publication:
2005
Abstract
It is shown that:
 a complete graph with a prime number p(>2) of
vertices can be properly edge-colored with p colors in
such a way that the edges can be partitioned into
edge-disjoint multicolored Hamitonian cycles.

When the number of vertices is n (≧8), with n a
power of two or five times a power of two, a proper
edge-coloring of the complete graph exists such that
its edges can be partitioned into isomorphic
multicolored spanning trees.
Basic terminology

Edge-disjoint :
two subgraphs are edge disjoint if
they do not share common edges.
e.g.
Basic terminology

Multicolored :
A graph with colored edges is called
multicolored if no two of its edges
have the same color.
e.g.
Basic terminology

Proper :
A coloring of edges of a graph is
proper if, whenever two edges have
one vertex in common, they carry
different colors.
e.g.
Basic terminology

unicycle :
A connected graph with m vertices and
m edges is called a unicycle.
e.g.
Background
A classical result of Euler,
that the edges of K2n can be partitioned
into isomorphic spanning trees (paths).
2
e.g. 1

4

3
we have two paths : 1-2-4-3 and 4-1-3-2 ,
They are isomorphic spanning paths in K4
Euler also decomposed K2n+1 into n
edge-disjoint Hamiltonian cycles.
Theorem (a)
For p(>2) prime there exists a proper
edge coloring of Kp that admits a
partition of edges into multicolored
Hamiltonian cycles.
1
Pf: 建立一個演算法.
2
5
e.g. 1
1
1

2
5
4
3
2
5
4
3
4
3
3
4
2
5
Theorem (b)
For n=2m, m≧3, or n=5*2m, m≧1,
there exists a proper edge coloring of
Kn that admits a partition of edges into
isomorphic multicolored spanning trees.
Pf: 略.
e.g.

1
2
6
1
3
5
4
2
6
3
5
4
Conjecture


Any proper coloring of the edges of a
complete graph on an odd number of
vertices allows a partition of the edges into
multicolored isomorphic unicyclic subgraphs.
Any proper coloring of the edges of a
complete graph on an even number (more
than four) of vertices allows a partition of
the edges into multicolored isomorphic
spanning trees.
Thanks for yours listening!
```