(4,5)-CYCLE SYSTEMS OF COMPLETE MULTIPARTITE GRAPHS

Abstract

In 1981, Alspach conjectured that if 3 <= m(i) <= v, v is odd and v(v - 1)/2 = m(1) +m(2)+ ... +m(t) then the complete graph K-v can be decomposed into t cycles of lengths m(1), m(2), ... , m(t) respectively; if v is even, v(v - 2)/2 = m(1) + m(2) ... + m(t), then the complete graph minus a one-factor K-v - F can be decomposed into t cycles of lengths m(1), m(2),..., Tat respectively. In this paper, we extend the study to the decomposition of the complete equipartite graph K-m(n).For m(i) is an element of {4, 5}, we prove that the trivial necessary conditions are also sufficient.