Weak colourings of Kirkman triple systems

Abstract

A $\delta$-colouring of the point set of a block design is said to be {\em weak} if no block is monochromatic. The {\em 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 $3(\mathrm{mod}6)$ there exists a Steiner triple system on $v$ points that has chromatic number $3$. Moreover, for each integer $\delta \ge 3$ there exist infinitely many Steiner triple systems with chromatic number $\delta$. We consider colourings of the subclass of Steiner triple systems which are resolvable. We show that for each $v\equiv 3(\mathrm{mod}6)$ there exists a Kirkman triple system on $v$ points with chromatic number $3$. We also show that for each integer $\delta \ge 3$, there exist infinitely many Kirkman triple systems with chromatic number $\delta$. We close with several open problems.