Various One-Factorizations of Complete Graphs

Methods to compute 1–factorizations of a complete graphs of even order are presented. For complete graphs where the number of vertices is a power of 2, we propose several new methods to construct 1–factorizations. Our methods are different from methods that make use of algebraic concepts such as Steiner triple systems, starters and all other existing methods. We also show that certain complete multipartite graphs have 1–factorizations by presenting a method to compute 1– factorizations of such graphs. This method can be applied to obtain 1–factorizations of complete graphs with the number of vertices being a multiple of 4 or complete graphs with mn vertices provided a 1–factorization of Km and a 1–factorization of Kn are known. Finally, deterministic and randomized back-tracking based algorithms to produce a 1–factorization for K2n are presented. Both the algorithms always produce a 1–factorization if one exists.