Directed graphs without rainbow triangles

TL;DR

This paper extends classical and colorful triangle-free graph problems to directed and oriented graphs, determining minimal edge conditions to avoid rainbow triangles across multiple colors.

Contribution

It provides the first solutions for directed and oriented graphs without rainbow triangles for any number of colors, with distinct cases for different triangle types.

Findings

Solved for directed graphs with any number of colors

Established results for transitive and oriented graphs

Different constructions and proofs for c=3 and c≥4

Abstract

One of the most fundamental results in graph theory is Mantel's theorem which determines the maximum number of edges in a triangle-free graph of order $n$. Recently a colorful variant of this problem has been solved. In such a variant we consider $c$ graphs on a common vertex set, thinking of each graph as edges in a distinct color, and want to determine the smallest number of edges in each color which guarantees existence of a rainbow triangle. Here, we solve the analogous problem for directed graphs without rainbow triangles, either directed or transitive, for any number of colors. The constructions and proofs essentially differ for $c=3$ and $c \geq 4$ and the type of the forbidden triangle. Additionally, we also solve the analogous problem in the setting of oriented graphs.