Bicolour graphs of Steiner triple systems

In this paper we introduce the notion of a bicolour graph of a Steiner triple system. Given an arbitrary colouring of the points of a Steiner triple system, the bicolour graph is the graph obtained on the point set of the system by placing an edge between two points if they have the same colour and appear in a triple with a point of a di4erent colour. The case where the graph is Kv was partially solved by Colbourn et al. (Electron. J. Combin. 6(1) (1999) 16) and the case where the graph is an independent set was solved by Milici et al. (Discrete Math. 240(1–3) (2001) 145–160). We show that every near 1-factor may occur as the bicolour graph of a Steiner triple system with four exceptions. In addition we consider the cases of a 2-factor and a near 2-factor and the complement of a 2-factor. c © 2002 Elsevier Science B.V. All rights reserved.