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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Another product construction for large sets of resolvable directed triple systems. | Another product construction for large sets of resolvable directed triple systems Hongtao Zhao School of Mathematics and Physics North China Electric Power University Beijing 102206 China ht_zhao@163.com Submitted Jul 26 2009 Accepted Sep 13 2009 Published Sep 18 2009 Mathematics Subject Classifications 05B07 Abstract A large set of resolvable directed triple systems of order v denoted by LRDTS v is a collection of 3 v 2 RDTS v s based on v-set X such that every transitive triple of X occurs as a block in exactly one of the 3 v 2 RDTS v s. In this paper we use DTRIQ and LR-design to present a new product construction for LRDTS v s. This provides some new infinite families of LRDTS v s. 1 Introduction Let X be a v-set. In what follows an ordered pair of X is always an ordered pair x y where x y E X. A transitive triple on X is a set of three ordered pairs x y y z and x z of X which is denoted by x y z . A directed triple system of order v denoted by DTS v is a pair X B where B is a collection of transitive triples on X called blocks such that each ordered pair of X occurs in exactly one block of B. A DTS v is called resolvable and is denoted by RDTS v if its blocks can be partitioned into subsets called parallel classes each containing every element of X exactly once. A large set of directed triple systems of order v denoted by LDTS v is a collection of 3 v 2 DTS v s based on X such that every transitive triple from X occurs as a block in exactly one of the 3 v 2 DTS v s. Existence results for LDTSs and RDTSs are well known from 1 9 . Theorem 1.1 1 There exists an LDTS v if and only if v 0 1 mod 3 and v 3. 2 There exists an RDTS v if and only if v 0 mod 3 v 3 and v 6. Research supported by NSFC Grant 10901051 NSFC Grant 10971051 and Doctoral Grant of North China Electric Power University. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R114 1 A large set of disjoint RDTS v s is denoted by LRDTS v . The existence of LRDTS v s has been investigated by Kang 8 Kang and
