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ALTERNATING COLOURINGS OF THE VERTICES OF A REGULAR POLYGON
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 100 / Issue 2 / October 2019
- Published online by Cambridge University Press:
- 13 February 2019, pp. 177-181
- Print publication:
- October 2019
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- Article
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Let
$n,r,k\in \mathbb{N}$. An 
$r$-colouring of the vertices of a regular 
$n$-gon is any mapping 
$\unicode[STIX]{x1D712}:\mathbb{Z}_{n}\rightarrow \{1,2,\ldots ,r\}$. Two colourings are equivalent if one of them can be obtained from another by a rotation of the polygon. An 
$r$-ary necklace of length 
$n$ is an equivalence class of 
$r$-colourings of 
$\mathbb{Z}_{n}$. We say that a colouring is 
$k$-alternating if all 
$k$ consecutive vertices have pairwise distinct colours. We compute the smallest number 
$r$ for which there exists a 
$k$-alternating 
$r$-colouring of 
$\mathbb{Z}_{n}$ and we count, for any 
$r$, 2-alternating 
$r$-colourings of 
$\mathbb{Z}_{n}$ and 2-alternating 
$r$-ary necklaces of length 
$n$.
COUNTING SYMMETRIC COLOURINGS OF THE VERTICES OF A REGULAR POLYGON
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 90 / Issue 1 / August 2014
- Published online by Cambridge University Press:
- 10 January 2014, pp. 1-8
- Print publication:
- August 2014
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- Article
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- You have access
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A colouring of the vertices of a regular polygon is symmetric if it is invariant under some reflection of the polygon. We count the number of symmetric
$r$-colourings of the vertices of a regular 
$n$-gon.
