Oberwolfach Partial Solution Of The Problem

The problem of determining whether Kn (for n odd) or Kn minus a1-factor (for n even) has an H-factorization is known as the Oberwolfach problem, which was formulated by Ringel at a graph theory meeting in1967. The notation OP(m1α1, m2α2,…, mtαt) represents the case in which each2-factor which is isomorphic to H consists of exactly αi cycles of length mi for1<i<t.Many results have been obtained in last decades. Let n be the number of vertices of graph G. The solution of OP(mt) has been solved for n=0(mod m). When n≠0(mod m), some people start to study whether there is a solution to OP(ma, sb). For the case of b=1, the resolution of s≥5ma-1has been solved, as well as some results on the resolution of1≤s≤5ma-2are obtained.In this paper, we shall prove the existence of a solution to OP(6a, s) and OP(6a, s2) for s∈{3,4,5,7,8}, and the existence of a solution to OP(5a, s2) for s∈{3,4,6,7}.