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Published online by Cambridge University Press: 13 February 2019
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 
$n$.
The first author was supported by NRF grant 107867 and the second author was supported by NRF grant IFR1202220164.
