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 Dn,s be the oriented graph such that V (Dn,s) is the set of integers mod 2n+1 and A(Dn,s) = {(i, j) : j − i ∈ {1, 2, . . . , n} \ {s}}. In this paper we prove that hc(Dn,s) ≤ 5 for n ≥ 7. The bound is tight since equality holds when s∈{n, 2n+1 3 }.