Resolvable Mendelsohn Triple Systems with Equal Sized Holes

An HMTS of type f n 1 ; n 2 ; \Delta \Delta \Delta ; n h g is a directed graph DK n 1 ;n 2 ;\Delta\Delta\Delta;n h which can be decomposed into 3-circuits. If the 3-circuits can be partitioned into parallel classes, then the HMTS is called an RHMTS. In this paper, it is shown that the RHMTSs of type m h exist when mh j 0 (mod 3) and (m; h) 6= (1; 6), with the possible exception of h = 6 and m62 M 17 , where M 17 = f m j m is divisible by a prime less than 17 g. The existence of Mendelsohn frames, which is closely related to RHMTS, is also considered in this paper. It is proved that a Mendelsohn frame of type t u exists if and only if u 4 and t(u \Gamma 1) j 0(mod 3) with 2 possible exceptions. Research supported by NSERC under grant A-5320 y Research supported by NSFC under grant 19231060-2 1 Introduction A complete multipartite directed graph DK n1 ;n2 ;\Delta\Delta\Delta;n h is a graph with vertex set X = S 1ih X i ; where X i (1 i h) are disjoint sets with jX i j = ..