Three-line chromatic indices of Steiner triple systems

There are five possible structures for a set of three lines of a Steiner triple system. Each of these three-line "configurations" gives rise to a colouring problem in which a partition of all the lines of an STS(v) is sought, the components of the partition each having the property of not containing any copy of the configuration in question. For a three-line configuration B, and STS (v) S, the minimum number of classes required is denoted by X(B, S) and is called the B-chromatic index of S. This generalises the ordinary chromatic index X'(S) and the 2-parallel chromatic index X′ (S). (For the latter see [7].) In this paper we obtain results concerning X(B, v) = min{x(B, S): S is an STS(v)} for four of the five three-line configurations B. In three of the cases we give precise values for all sufficiently large v and in the fourth case we give an asymptotic result. The values of the four chromatic indices for v≤13 are also determined.