Cycle frames and the Oberwolfach problem

For an integer λ, G(λ) denotes a graph G with uniform edge-multiplicity λ. Let J be a subset of positive integers. A 2-regular subgraph of m-partite graph Km⊗In containing vertices of all but one partite set is called partial 2-factor, where ⊗ denotes wreath product and In is an independent set on n vertices. If (Km⊗In)(λ) can be partitioned into edge-disjoint partial 2-factors such that each partial 2-factor consists of cycles of lengths from J, then we say that (Km⊗In)(λ) has a (J,λ)-cycle frame. The Oberwolfach problem OP (m1α1,m2α2,…,mtαt), raised by Ringel, asks the existence of a 2-factorization of Kn (when n is odd) or Kn−I (when n is even), in which each 2-factor consists of exactly αi cycles of length mi, i=1,2,…,t. In this paper, we show that there exists a ( 4,6 ,λ)-cycle frame of (Km⊗In)(λ) if and only if λn≡0(mod2), m≥3, (n,m)∉ (1,3),(2t+1,2s)∣t≥0,s≥2 . Further we show that there exists a ( 3,4 ,1)-cycle frame of Km⊗In if and only if m≥3 and n≡0(mod2). As a consequence, we solve OP(3a,4b), OP(3a,6b) and OP(5a,10b) with some restrictions on a,b∈N.

Abstract licence: CC BY-NC-ND

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