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Existence of 3-chromatic Steiner quadruple systems

✍ Scribed by L. Ji

Publisher: John Wiley and Sons
Year: 2007
Tongue: English
Weight: 139 KB
Volume: 15
Category: Article
ISSN: 1063-8539
DOI: 10.1002/jcd.20125

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✦ Synopsis

Abstract

A Steiner quadruple system of order v (briefly SQS (v)) is a pair (X, $\cal B$), where X is a v‐element set and $\cal B$ is a set of 4‐element subsets of X (called blocks or quadruples), such that each 3‐element subset of X is contained in a unique block of $\cal B$. The chromatic number of an SQS(v)(X, $\cal B$) is the smallest m for which there is a map $\varphi : X \rightarrow Z_m$ such that $|\varphi(B)|\geq 2$ for all $B \in \cal B$, where $\varphi (B) ={\varphi (x):x\in B}$. The system (X, $\cal B$) is equitably m‐chromatic if there is a proper coloring $\varphi$ with minimal m for which the numbers $|\varphi^{-1}(c)|, c\in Z_m$ differ from each other by at most 1. Linek and Mendelsohn showed that an equitably 3‐chromatic SQS(v) exists for v ≡ 4, 8, 10 (mod 12), v ≥ 16. In this article we show that an equitably 3‐chromatic SQS(v) exists for v ≡ 2 (mod 12) with v > 2. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 469–477, 2007

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