Weak colourings of Kirkman triple systems
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910912175144960 |
|---|---|
| author | Burgess, Andrea C. Cavenagh, Nicholas J. Danziger, Peter Pike, David A. |
| author_facet | Burgess, Andrea C. Cavenagh, Nicholas J. Danziger, Peter Pike, David A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06066 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak colourings of Kirkman triple systems Burgess, Andrea C. Cavenagh, Nicholas J. Danziger, Peter Pike, David A. Combinatorics 05B07, 05B30, 05C15 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. |
| title | Weak colourings of Kirkman triple systems |
| topic | Combinatorics 05B07, 05B30, 05C15 |
| url | https://arxiv.org/abs/2404.06066 |