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Abstract. For an even integer m at least 4 and any positive integer t, itis shown that the complete equipartite graph K km(2t) can be decomposedinto edge-disjoint gregarious m-cycles for any positive integer k under thecondition satisfying (m 1) 2 +34m < k. Here it will be called a gregariouscycle if the cycle has at most one vertex from each partite set. 1. IntroductionA complete multipartite graph with partite sets of order a 1 ;a 2 ;:::;a n willbe denoted by K(a 1 ;a 2 ;:::;a n ) so that any two vertices in the same partiteset have no edge joining them while any two vertices in di erent partite setshave one edge joining them. If all the partite sets have the same size of ordermthen we refer to the complete multipartite graph as equipartite, and use thenotation K n(m) to denote this graph with nparts of size m.The problem of the existence of edge-disjoint decompositions of completegraphs and complete multipartite graphs into cycles of a xed length havebeen considered in a number of di erent ways. After a series of developments,the problem of decomposing a complete graph of odd order, and also a com-plete graph of even order minus 1-factor into cycles of a xed length has beencompletely solved (see [1], [11] and [12]). But it is still an open problem whenK

ON DECOMPOSITIONS OF THE COMPLETE EQUIPARTITE GRAPHS Kkm(2t)INTO GREGARIOUS m-CYCLES

ON DECOMPOSITIONS OF THE COMPLETE EQUIPARTITE GRAPHS K_km(2t)INTO GREGARIOUS m-CYCLES