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arXiv · 2404.06066

Weak colourings of Kirkman triple systems

Andrea C. Burgess·Nicholas J. Cavenagh·Peter Danziger·David A. Pike
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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 $3\pmod{6}$ there exists a Steiner triple system on $v$ points that has chromatic number $3$. Moreover, for each integer $\delta \geq 3$ 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 $v\equiv 3\pmod{6}$ there exists a Kirkman triple system on $v$ points with chromatic number $3$. We also show that for each integer $\delta \geq 3$, there exist infinitely many Kirkman triple systems with chromatic number $\delta$. We close with several open problems.

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Andrea C. Burgess, Nicholas J. Cavenagh, Peter Danziger, David A. Pike. 2024-04-09. Weak colourings of Kirkman triple systems. https://arxiv.org/abs/2404.06066

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