Uniquely 3-colourable Steiner triple systems
type: Item
instance of: scholarly article
title: Uniquely 3-colourable Steiner triple systems
zbMATH Open document ID: 1012.05030
DOI: 10.1016/S0097-3165(02)00016-X
published in: Journal of Combinatorial Theory. Series A
publication date: 2003-03-26T00:00:00Z
review text: A (weak) 3-colouring of a Steiner triple system \(S\) of order \(v\) (or STS(\(v\))) is an assignment of colours from a set of cardinality 3 to the \(v\) points of \(S\) in such a way that every block contains at least two points with different colours. A 3-colouring is equitable if the cardinalities of the sets of points assigned each colour differ by at most one. If every 3-colouring of \(S\) is equitable then \(S\) is called 3-balanced. A 3-colouring of \(S\) is unique if every other 3-colouring can be obtained from the original colour assignment by a permutation of the colours. The authors define a type-I STS(\(v\)) as a 3-balanced STS(\(v\)) with a unique (necessarily equitable) 3-colouring, and a type-II STS(\(v\)) as an STS(\(v\)) with a unique 3-colouring which is non-equitable. They show that a type-I STS(\(v\)) exists for every admissible \(v \geq 25\). Since it is known that no type-I STS(\(v\)) exists for any \(v \leq 15\), the only remaining undecided cases are \(v = 19\) and \(21\). In addition they show that a type-II STS(\(v\)) exists for every admissible \(v \geq 25\). Since it is known that no type-II STS(\(v\)) exists for any \(v \leq 19\), the only remaining undecided case is \(v = 21\). Group divisible designs with block size 3 and Wilson's fundamental construction are important ingredients in the recursive construction results.
reviewed by: Peter B. Gibbons
Mathematics Subject Classification ID: 05B07
zbMATH DE Number: 1888438
zbMATH Keywords: Steiner triple system
zbMATH Keywords: 3-colouring
author: Anthony D. Forbes
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1888438