Rainbow triangles in arc-colored digraphs

TL;DR

This paper determines the minimum number of colors needed to guarantee a rainbow triangle in arc-colored complete digraphs and tournaments, providing characterizations and extending known undirected results.

Contribution

It explicitly calculates the threshold function f(D) for complete digraphs and strongly connected tournaments, and characterizes colorings avoiding rainbow triangles at this threshold.

Findings

Calculated f(\overleftrightarrow{K}_{n}) and f(T_n) for complete digraphs and tournaments.

Characterized colorings of complete digraphs with no rainbow triangles at the threshold.

Proved a new bound involving arc and color counts guaranteeing rainbow triangles.

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 $c(D)\geq f(D)$, then $D$ contains a rainbow triangle. In this paper we obtain $f(\overleftrightarrow{K}_{n})$ and $f(T_n)$, where $\overleftrightarrow{K}_{n}$ 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 digraph $\overleftrightarrow{K}_{n}$ with $c(\overleftrightarrow{K}_{n})=f(\overleftrightarrow{K}_{n})-1$ and containing no rainbow triangles. We also prove that an arc-colored digraph $D$ on $n$ vertices contains a rainbow triangle when $a(D)+c(D)\geq a(\overleftrightarrow{K}_{n})+f(\overleftrightarrow{K}_{n})$, which is a directed extension of the undirected case.