Decomposing Steiner Triple Systems Into Four-Line Configurations

Abstract

Publisher Summary Decomposing Steiner Triple Systems Into Four-Line ConfigurationsThis chapter discusses decompositions of Steiner triple system (STS) into four-line configurations. A Steiner triple system (STS) is a pair (V, B) where V is a v-set of elements, and B is a collection of three-subsets of V called “triples” or “lines” such that every two-subset of V is contained in exactly one triple. The number v = | V | is called the “order” of the STS. An STS of order v (STS(v)) exists if and only if v ≡ 1 or 3 (mod 6). If in the definition of an STS “exactly” is replaced with “at most,” one has a “partial triple system.” The chapter uses the term “configuration” to describe a partial triple system with a fixed small number of lines. It discusses only “exact” decomposition—that is, the chapter is restricted to STSs with a set of triples B such that 4 divides | B |, the cardinality of B .