Decomposition of bipartite graphs into special subgraphs

Let F and G be two graphs and let H be a subgraph of G. A decomposition of G into subgraphs image is called an F-factorization of G orthogonal to H if image and image for each image. Gyárfás and Schelp conjectured that the complete bipartite graph image has a image-factorization orthogonal to H provided that H is a k-factor of image. In this paper, we show that (1) the conjecture is true when H satisfies some structural conditions; (2) for any two positive integers image, image has a image-factorization orthogonal to H if H is a k-factor of image; (3) image has a image-factorization such that each edge of H belongs to a different image if H is a subgraph of image with maximum degree image.