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Cyclic antiautomorphisms of directed triple systems

✍ Scribed by Neil P. Carnes; Anne Dye; James F. Reed

Publisher: John Wiley and Sons
Year: 1996
Tongue: English
Weight: 645 KB
Volume: 4
Category: Article
ISSN: 1063-8539
DOI: 10.1002/(sici)1520-6610(1996)4:2<105::aid-jcd3>3.0.co;2-j

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✦ Synopsis

A transitive triple, (a,b,c), is defined to be the set )} of ordered pairs. A directed triple system of order v, DTS(v), is a pair (D, p), where D is a set of v points and fi is a collection of transitive triples of pairwise distinct points of D such that any ordered pair of distinct points of D is contained in precisely one transitive triple of /3. An antiautomorphism of a Directed triple system, ( D , / 3 ) , is a permutation of D that maps / 3 to /3-', where

~-' = { ( c . b , a ) ~( a , b , c ) € f i } .

In this article we give necessary and sufficient conditions for the existence of a Directed triple system of order v admitting an antiautomorphism consisting of a single cycle of length d and having vd fixed points. Further, we give a more general result for partial Directed triple systems in which the missing ordered pairs are precisely those containing two fixed points.

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