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 have 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 (T n ), where ← → K n is a complete digraph of order n and T n is a strongly connected tournament of order n. Moreover we characterize the arc-colored complete digraph1 and containing no rainbow triangles. We also prove that an arc-colored digraph D on n vertices contains a rainbow triangle when, which is a directed extension of the undirected case.