| An (m1,m2,... .mt) - 2-factorization of G is a 2-factorization of which each 2-factor consists of cycles of lengths m1, m2 , ..., mt, The problem of determinning whether there exists an (m1, m2, ..., mt) - 2- factorization of Kv, when v is odd, is the Oberwolfach problem, denoted OP(m1,m2,...,mt), Moreover, we denote a 2-factorization of Kv by OP(v: {3. t*}) which satisfies that all cycles are of length three except that each 2-factor contains one cycle of length t. The triangle spectrum for 2-factorizations of the complete graph Kv is the set (v) ={ |there exists a 2-factorizations of Kv in which the total number of triangles equals }.Dejter, Frannek. Mendelsohn and Rosa [6] determined the triangle spectrum for all v = 1,3 (mod 6), and formulate an open problem.Open problem: determine the triangle spectrum for v = 5 (mod 6).In this paper, we solve the open problem, and give the existence for OP(v; {3, 5*})simultaneously,... |
