Weak colourings of Kirkman triple systems

Abstract

A \(\delta \)-colouring of the point set of a block design is said to be weak if no block is monochromatic. The chromatic number \(\chi (S)\) of a block design S is the smallest integer \(\delta \) such that S has a weak \(\delta \)-colouring. It has previously been shown that any Steiner triple system has chromatic number at least 3 and that for each \(v\equiv 1\) or \(3\pmod {6}\) there exists a Steiner triple system on v points that has chromatic number 3. Moreover, for each integer \(\delta \geqslant 3\) there exist infinitely many Steiner triple systems with chromatic number \(\delta \). We consider colourings of the subclass of Steiner triple systems which are resolvable. A Kirkman triple system consists of a resolvable Steiner triple system together with a partition of its blocks into parallel classes. We show that for each \(v\equiv 3\pmod {6}\) there exists a Kirkman triple system on v points with chromatic number 3. We also show that for each integer \(\delta \geqslant 3\), there exist infinitely many Kirkman triple systems with chromatic number \(\delta \). We close with several open problems.