EDGE-DISJOINT ISOMORPHIC MULTICOLORED TREES AND CYCLES IN COMPLETE GRAPHS 應數100 9622053 吳家寶 Abstract    Author Gregory M. Constantine Publisher Society for Industrial and Applied Mathematics Philadelphia, PA, USA 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!