1. EDGE-DISJOINT ISOMORPHIC MULTICOLORED TREES AND CYCLES IN COMPLETE GRAPHS 應數 100 9622053 吳家寶

2. Abstract Author Gregory M. Constantine Publisher Society for Industrial and Applied Mathematics Philadelphia, PA, USA Year of Publication: 2005

3. 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.

4. Basic terminology Edge-disjoint : two subgraphs are edge disjoint if they do not share common edges. e.g.

5. Basic terminology Multicolored : A graph with colored edges is called multicolored if no two of its edges have the same color. e.g.

6. 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.

7. Basic terminology unicycle : A connected graph with m vertices and m edges is called a unicycle. e.g.

8. Background A classical result of Euler, that the edges of K 2n can be partitioned into isomorphic spanning trees (paths). e.g. Euler also decomposed K 2n+1 into n edge-disjoint Hamiltonian cycles. 12 34 we have two paths : 1-2-4-3 and 4-1-3-2, They are isomorphic spanning paths in K 4

9. Theorem (a) For p(>2) prime there exists a proper edge coloring of K p that admits a partition of edges into multicolored Hamiltonian cycles. Pf: 建立一個演算法. e.g. 1 4 2 3 5 1 4 2 3 5 1 2 3 5 4 1 4 2 3 5

10. Theorem (b) For n=2 m, m ≧ 3, or n=5*2 m, m ≧ 1, there exists a proper edge coloring of K n that admits a partition of edges into isomorphic multicolored spanning trees. Pf: 略. e.g. 1 6 5 2 3 4 1 6 5 2 3 4

11. 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.

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