Weak colourings of Kirkman triple systems

TL;DR

This work establishes that every admissible order $v ≡ 3 mod 6$ admits a 3-chromatic Kirkman triple system (KTS$(v)$) and proves the existence of infinitely many KTS with any prescribed chromatic number $oldsymbol{δ} ≥ 3$ using frame-based and quadruple-system constructions. Central to the approach are rainbow colorings, Kirkman frames, and group-divisible designs that enable lifting local rainbow colorings to larger resolvable designs, along with K(Q) constructions from quadruple systems that relate the chromatic number of the base and the resulting KTS. The paper also provides concrete results for small orders, including explicit rainbow colorings for several $v$ and a 4-chromatic family of KTS$(32u+1)$ for certain $u$, while outlining (and experimentally exploring) open problems about higher chromatic numbers and optimal bounds. Collectively, these contributions expand the catalog of KTS with prescribed chromatic properties and highlight fertile directions for extending colorability results to broader resolvable BIBDs and related combinatorial designs.

Abstract

A $δ$-colouring of the point set of a block design is said to be {\em weak} if no block is monochromatic. The {\em chromatic number} $χ(S)$ of a block design $S$ is the smallest integer $δ$ such that $S$ has a weak $δ$-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 $δ\geq 3$ there exist infinitely many Steiner triple systems with chromatic number $δ$. 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 $δ\geq 3$, there exist infinitely many Kirkman triple systems with chromatic number $δ$. We close with several open problems.

Theorems & Definitions (45)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Remark 3.1