On the oberwolfach problem OP(3a, sb)

Abstract

The problem of determining whether K_n(for n odd) or K _n minus a 1-factor (for n even) has a H-factorization isknown as the Oberwolfach problem.The notation OP(m^α1₁,m ^α2₂,⋯,m^αt_t) represents the case in which each 2-factor must consist of exactly α_i m_i-cycles for i = 1,2,⋯, t.In this paper, we shall mainly prove that the necessary condition for the existence of 0P(3^a,s^b) with 1 < b < 3 and s = 4,5 is also sufficient with one exception.