image-factorization of complete bipartite graphs Original Research Article

A image-factor of complete bipartite graph image is a spanning subgraph of image such that every component is a path of length k. A image-factorization of image is a set of edge-disjoint image-factors of image which is a partition of the set of edges of image. When k is an even number, the spectrum problem for a image-factorization of image has been completely solved. When k is an odd number, Ushio in 1993 proposed a conjecture. However, up to now we only know that Ushio Conjecture is true for image. In this paper we will show that Ushio Conjecture is true when image. That is, we shall prove that a necessary and sufficient condition for the existence of a image-factorization of image is (1) image, (2) image, (3) image (mod 5), and (4) image is an integer.