The chromatic index of block intersection graphs of Kirkman triple systems and cyclic Steiner triple systems

The block intersection graph of a combinatorial design with block set B is the graph with B as its vertex set such that two vertices are adjacent if and only if their associated blocks are not disjoint. The chromatic index of a graph G is the least number of colours that enable each edge of G to be assigned a single colour such that adjacent edges never have the same colour. A graph G for which the chromatic index equals the maximum degree is called Class 1; otherwise the chromatic index exceeds the maximum degree by one and G is called Class 2. We conjecture that whenever a Steiner triple system has a block intersection graph with an even number vertices, the graph is Class 1. We prove this to be true for Kirkman triple systems and cyclic Steiner triple systems of order v ≡ 9 (mod 12). We also prove that the conjecture holds for cyclic Steiner triple systems of order v ≡ 1 (mod 12) for which φ(v) v−1 2 3 , where φ is Euler’s totient function.