• decomposition;
  • factorization;
  • cycle frame


For two graphs G and H their wreath product inline image has vertex set inline image in which two vertices inline image and inline image are adjacent whenever inline image or inline image and inline image. Clearly, inline image, where inline image is an independent set on n vertices, is isomorphic to the complete m-partite graph in which each partite set has exactly n vertices. A 2-regular subgraph of the complete multipartite graph inline image containing vertices of all but one partite set is called partial 2-factor. For an integer λ, inline image denotes a graph G with uniform edge multiplicity λ. Let J be a set of integers. If inline image can be partitioned into edge-disjoint partial 2-factors consisting cycles of lengths from J, then we say that inline image has a inline image-cycle frame. In this paper, we show that for inline image and inline image, there exists a inline image-cycle frame of inline image if and only if inline image and inline image. In fact our results completely solve the existence of a inline image-cycle frame of inline image.