Rainbow triangles in arc-colored digraphs

Abstract

Let D be an arc-colored digraph. The arc number a(D) of D is defined as the number of arcs of D. The color number c(D) of D is defined as the number of colors assigned to the arcs of D. A rainbow triangle in D is a directed triangle in which every pair of arcs has distinct colors. Let f(D) be the smallest integer such that if c(D)≥f(D), then D contains a rainbow triangle. In this paper we obtain f(K↔n) and f(Tn), where K↔n is a complete digraph of order n and Tn is a strongly connected tournament of order n. Moreover we characterize the arc-colored complete digraph K↔n with c(K↔n)=f(K↔n)−1 and containing no rainbow triangles. We also prove that an arc-colored digraph D on n vertices contains a rainbow triangle when a(D)+c(D)≥a(K↔n)+f(K↔n), which is a directed extension of the undirected case.