Holey Perfect Mendelsohn Designs of Type with Block Size Four

Let 4-HPMD denote a holey perfect Mendelsohn design with block size four. The existence of 4-HPMDs with holes of size 2 and one hole of size 3, that is, of type , was established by Bennett et al. in 1997. In this paper, we investigate the existence of 4-HPMDs of type for : a 4-HPMD( ) exists if and only if , except possibly for , (7, 6), (11, 9), (11, 10). We also investigate the existence of 4-HPMD( ) for general and prove that there exists a 4-HPMD( ) for all . Moreover, if , then a 4-HPMD( ) exists for all ; if , then a 4-HPMD( ) exists for all . 1. Introduction Let be positive integers, a -Mendelsohn design, briefly -MD, is a pair , where is a -set (of points) and is a collection of cyclically ordered -subset of (called blocks) such that every ordered pair of points of are consecutive in exactly blocks of . If for all , every ordered pair of points of is -apart in exactly blocks of , then the -MD is called perfect and is denoted by -PMD. The existence of a -PMD is equivalent to the existence of an idempotent quasigroup satisfying the identity, called Stein's third law, for all . Let . Then is a -PMD. For the existence of -PMD, we have the following theorem from [1–3]. Theorem 1. A -PMD exists if and only if and . In this paper, we are only interested in PMDs where . A holey perfect Mendelsohn design (briefly HPMD) is a triple which satisfies the following properties. (1) is a partition of into subsets called holes. (2) is a family of cyclically ordered -subset of (called blocks) such that a hole and a block contain at most one common point.(3)Every ordered pair of points from distinct holes are -apart in exactly one block for . The type of the HPMD is the multiset and is described by an exponential notation. Throughout this paper, we shall use HPMD to describe a PMD of the type in which occurs times, . In graph theoretic terminology, an HPMD is a decomposition of a complete multipartite directed graph DK into -circuits such that for any two vertices and from different components, there is one circuit along which the directed distance from to is , where . Another class of designs related to HPMDs is group divisible design (GDD). A GDD is a 4-tuple ) which satisfies the following properties. (1) is a partition of into subsets called groups. (2) is a family of subsets of (called blocks) such that a group and a block contain at most one common point. (3)Every pair of points from distinct groups occurs in exactly blocks. The type of the GDD is the multiset and we will also use an “exponential” notation for the type of GDD. We also use the notation GDD(