Alternating Colourings of the Vertices of a Regular Polygon

Abstract

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 $n$.

Enumerating Necklaces With Transitions

“…In this case, there cannot be any fixed elements when p d, where p is the period of the composition. Otherwise, when p | d, for a proper coloured composition to be fixed, all the parts belonging to the same cycle need to have the same size and colour, and the colours of two adjacent cycles need to be different (Figure 6 We now show how the previously known solutions for t = 2 and t = n (Section 3.2) can be easily obtained as special cases of Theorem The case t = n corresponds to counting the number of necklaces in n beads and k colours with no adjacent beads having the same colours, or, equivalently, the number of alternating colourings of the vertices of a regular n-agon, which was recently treated in [13]. By the definition of a p-periodic cyclic composition,…”

“…The case t = n (colourful necklaces [4]) is tantamount to counting the number of alternating colourings of the vertices of a regular polygon, two colourings being equivalent if they can be obtained from one another by a rotation of the polygon (Figure 4). The problem was recently investigated by Singh and Zelenyuk [13] who obtained…”

“…of n into t parts and let Ã[S k (n, t)] denote the orbits of A[S k (n, t)] under the action of C t . We proceed by Burnside's lemma [3,Theorem 8.4.3], extending the approach described in [13]. Start by recalling that the cycle index of the cyclic group C t is In the remainder of this proof, for convenience, we shall also represent compositions of n into t parts as cyclic graphs on t vertices, with each vertex corresponding to one part of the composition (see Figure 6).…”