TAILIEUCHUNG - Báo cáo toán hoc:"Another product construction for large sets of resolvable directed triple systems"

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@ 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 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

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