The spectrum of equitable, 4‐chromatic Steiner quadruple systems

Abstract

AbstractA Steiner quadruple system of order v (briefly an SQS(v)) is a pair (X,$\cal B$) with |X| = v and $\cal B$ a set of quadruples taken from X such that every triple in X is in a unique quadruple in $\cal B$. Hanani [Canad J Math 12 (1960), 145–157] showed that an SQS(v) exists if and only if v is {admissible}, that is, v = 0,1 or v ≡ 2,4 (mod 6). Each SQS(v) has a chromatic number when considered as a 4‐uniform hypergraph. Here we show that a 4‐chromatic SQS(v) exists for all admissible v ≥ 20, and that no 4‐chromatic SQS(v) exists for v < 20. Each system we construct admits a proper 4‐coloring that is equitable, that is, any two color classes differ in size by at most one. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 369–392, 2007