Disproof of a conjecture on the rainbow triangles in arc-colored digraphs

Abstract

Let D be a digraph. The arc number a(D) of D is defined as the number of arcs of D. An arc-coloring of D is a mapping C: A(D)→N, where N is the set of natural numbers. 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. Recently, Li et al. (2022) proved that if a(D)+c(D)=n(n−1)+⌊n24⌋+2 for an arc-colored digraph D of order n=3,4 without containing rainbow triangles, then D≅Kn↔. Thus, they conjectured that if a(D)+c(D)=n(n−1)+⌊n24⌋+1 for an arc-colored digraph D of order n≥5 without containing rainbow triangles, then D≅Kn↔. In this note, we disprove the conjecture by constructing a family of counterexamples. In fact, we show that there is not a function f(n) such that D≅Kn↔ if a(D)+c(D)=f(n) for an arc-colored digraph D of order n≥5 without containing rainbow triangles.