Digital Object Identifier (DOI) 10.1007/s00373-008-0794-8 Graphs and Combinatorics (2008) 24:313–326 Graphs and Combinatorics © Springer-Verlag 2008 Equitable Specialized Block-Colourings for Steiner Triple Systems Mario Gionfriddo 1 , Peter Hor´ ak 2 , Lorenzo Milazzo 1 , Alex Rosa 3 1 University of Catania, Catania, Italy. 2 University of Washington, Tacoma, WA, USA. 3 McMaster University, Hamilton, ON, Canada. Abstract. We continue the study of specialized block-colourings of Steiner triple systems initiated in [2] in which the triples through any element are coloured according to a given partition π of the replication number. Such colourings are equitable if π is an equitable par- tition (i.e., the difference between any two parts of π is at most one). Our main results deal with colourings according to equitable partitions into two, and three parts, respectively. Key words. Steiner triple systems, block-colourings, equitable. 1. Introduction A Steiner triple system of order v (STS(v)) is a pair (V , B) where V is v-set of ele- ments and B is a family of 3-subsets of V called triples such that every 2-subset of V is contained in exactly one triple of B. It is well known that an STS(v) exists if and only if v ≡ 1 or 3 (mod 6) [1]. Every element of an STS(V ) is contained in r = v-1 2 triples; r is called the replication number. An STS(v) (V , B) is cyclic if it admits an automorphism α consisting of a single cycle of length v which preserves B. A block-colouring of an STS(v) (V , B) is a mapping φ : B → C where C is a set of colours. A k -block-colouring (or simply a k -colouring) is a block-colouring using k colours; each of the k colours must be used. For each i = 1,..., k , the subset B i of B containing the blocks coloured with colour i is a colour class. For a partition π ={π 1 ,π 2 ,...,π s } of the replication number r ,a k -colouring of type π is a colouring of triples such that for each element v ∈ V , the triples con- taining v are partitioned according to π , that is, there are π 1 triples of one colour, π 2 triples of a different colour, and so on. For an STS(v) S = (V , B) and a partition π of r , we define the colour spectrum  π ( S) ={k : there exists a k -block-colouring of type π of S}, and also define  π (v) =∪ π ( S) where π is a partition of r into s parts, s > 1, and where the union is taken over the set of all STS(v). From now on, we will simply write k -colouring instead of k -block-colouring since only block-colourings will be considered.