On the heterochromatic number of circulant digraphs

The heterochromatic number hc(D) of a digraph D, is the minimum integer k such that for every partition of V(D) into k classes, there is a cyclic triangle whose three vertices belong to different classes. For any two integers s and n with 1 ≤ s ≤ n, let ${\displaystyle D_{n,s}}$ be the oriented graph such that ${\displaystyle V\left(D_{n,s}\right)}$ is the set of integers mod 2n+1 and ${\displaystyle A\left(D_{n,s}\right)=\{(i,j):j-i\in \{1,2,...,n\}\setminus \left\{s\right\}\}..Inthispaperweprovet\hat{}}$hc(D_{n,s}) ≤ 5!$! for n ≥ 7. The bound is tight since equality holds when s ∈ {n,[(2n+1)/3]}.