The spectrum of resolvable holey Mendelsohn triple systems and holey Mendelsohn frames with block size three

A holey Mendelsohn triple system (HMTS) is a decomposition of a complete multipartite directed graph into directed cycles of length 3. If the directed cycles of length 3 can be partitioned into parallel classes, then the HMTS is called an RHMTS. Bennett, Wei and Zhu [J. Combin. Des., 1997] showed that an RHMTS of type gn exists when gn≡0(mod3) and (g,n)≠(1,6) with some possible exceptions. In this paper, motivated by the application in constructing RHMTSs, we investigate the constructions of holey Mendelsohn frames. We prove that a 3-MHF of type (n,ht) exists if and only if n≥3, t≥4 and nh(t−1)≡0(mod3), and then determine that the necessary condition for the existence of an RHMTS of type gn, namely, gn≡0(mod3) is also sufficient except for (g,n)=(1,6). New recursive constructions on incomplete RHMTSs via MHFs are introduced to settle this problem completely.