(4, 5)-cycle systems of complete multipartite graphs

In 1981, Alspach conjectured that if 3 ≤ m_i ≤ v, v is odd and v(v-1)/2 = m₁+m₂+···+m_t, then the complete graph K_v can be decomposed into t cycles of lengths m₁,m₂,···,m_t respectively; if v is even, v(v-2)/2 = m₁+m₂+···+m_n, then the complete graph minus a one-factor K_v - F can be decomposed into t cycles of lengths m₁,m₂,···,m_n respectively. In this paper, we extend the study to the decomposition of the complete equipartite graph K_m(n). For m_i ∈ {4, 5}, we prove that the trivial necessary conditions are also sufficient.