On large sets of resolvable and almost resolvable oriented triple systems

✍ Scribed by Qingde Kang; Jianguo Lei

• Publisher: John Wiley and Sons
• Year: 1996
• Tongue: English
• Weight: 430 KB
• Volume: 4
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/(sici)1520-6610(1996)4:2<95::aid-jcd2>3.0.co;2-u

✦ Synopsis

An MTS(v) [or DTS(v)] is said to be resolvable, denoted by RMTS(v) [or RDTS(v)], if its block set can be partitioned into parallel classes. An MTS(v) [or DTS(v)] is said to be almost resolvable, denoted by ARMTS(v) [or ARDTS(v)], if its bloak set can be partitioned into almost parallel classes. The large set of RMTS(v) [or RDTS(v) or ARMTS(v) or ARDTS(v)] is denoted by LRMTS(v) [or LRDTS(v) or LARMTS(v) or LARDTS(v)]. In this article we do some preliminary study for their existence, and give several recursive theorems using other combinatorial structures. 0 1996 John Wiley & Sons, Inc.

1. Introduction

Let X be a finite set. In what follows an orderedpair of X will always be an ordered pair (x, y ) where x # y E X . A cyclic triple on X is a set of three ordered pairs (x, y ) , ( y , z ) , and ( z , x) of X , which is denoted by (x, y , z ) (or ( y , z , x) or ( z , x, y ) ) . A transitive triple on X is a set of three ordered pairs (x,y), ( y , z ) , and ( x ~) of X, which is denoted by An oriented triple systems of order v , briefly OTS(v), is a pair ( X , 3) where X is a v-set and Z? is a collection of cyclic or transitive triples on X, called blocks, such that every ordered pair of X belongs to exactly one block of 3. In particular, if the triples in B are all cyclic (or transitive) then (X, B ) is called Mendelsohn (or directed) triple systems and denoted by MTS(v) (or DTS(v)).

For a v-set X, some cyclic (or transitive) triples on X are said to be a parallel class if they form a partition of X. Some cyclic (or transitive) triples on X are said to be an (X,Y>Z).