Cycle frames and the Oberwolfach problem

Abstract

For an integer denotes a graph with uniform edge-multiplicity Let be a subset of positive integers. A 2-regular subgraph of -partite graph containing vertices of all but one partite set is called partial 2-factor, where denotes wreath product and is an independent set on vertices. If can be partitioned into edge-disjoint partial 2-factors such that each partial 2-factor consists of cycles of lengths from then we say that has a -cycle frame. The Oberwolfach problem OP raised by Ringel, asks the existence of a 2-factorization of (when is odd) or (when is even), in which each 2-factor consists of exactly cycles of length , In this paper, we show that there exists a -cycle frame of if and only if , , . Further we show that there exists a -cycle frame of if and only if and As a consequence, we solve OP OP and OP with some restrictions on

Keywords

• decomposition
• factorization
• cycle frame