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arXiv · 2308.01461

Directed graphs without rainbow triangles

Sebastian Babiński·Andrzej Grzesik·Magdalena Prorok
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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.

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Sebastian Babiński, Andrzej Grzesik, Magdalena Prorok. 2023-08-05. Directed graphs without rainbow triangles. https://arxiv.org/abs/2308.01461

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