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Steiner triple systems with two disjoint subsystems

✍ Scribed by Darryn Bryant; Daniel Horsley

Publisher: John Wiley and Sons
Year: 2005
Tongue: English
Weight: 123 KB
Volume: 14
Category: Article
ISSN: 1063-8539
DOI: 10.1002/jcd.20071

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✦ Synopsis

Abstract

It is shown that there exists a triangle decomposition of the graph obtained from the complete graph of order v by removing the edges of two vertex disjoint complete subgraphs of orders u and w if and only if u,w, and v are odd, ${{v}\choose 2}-{u\choose 2}- {w\choose 2}\equiv 0$ (mod 3), and ${v}\ge w + u +{\rm max} { u,w}$. Such decompositions are equivalent to group divisible designs with block size 3, one group of size u, one group of size w, and v – u – w groups of size 1. This result settles the existence problem for Steiner triple systems having two disjoint specified subsystems, thereby generalizing the well‐known theorem of Doyen and Wilson on the existence of Steiner triple systems with a single specified subsystem. © 2005 Wiley Periodicals, Inc. J Combin Designs

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