A note on cyclic m-cycle systems of K-r((m))

Abstract

It was proved by Buratti and Del Fra that for each pair of odd integers r and m, there exists a cyclic m-cycle system of the balanced complete r-partite graph K-r(m) except for the case when r=m=3. In this note, we study the existence of a cyclic m-cycle system of K-r(m) where r or m is even. Combining the work of Buratti and Del Fra, we prove that cyclic m-cycle systems of K-r(m) exist if and only if (a) K-r(m) is an even graph (b) (r, m) not equal (3, 3) and (c) (r,m) not equivalent to (t , 2) (mod 4) where t is an element of {2,3}.