Circulant tournaments of prime order are tight

AbstractWe say that a tournament is tight if for every proper 3-coloring of its vertex set there is a directed cyclic triangle whose vertices have different colors. In this paper, we prove that all circulant tournaments with a prime number p≥3 of vertices are tight using results relating to the acyclic disconnection of a digraph and theorems of additive number theory

Avances en Matemática Discreta en Andalucía. V Encuentro andaluz de Matemática Discreta. La Línea de la Concepción (Cádiz), 4-5 de julio de 2007

V Encuentro andaluz de Matemática Discreta. La Línea de la Concepción (Cádiz), 4-5 de julio de 200

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