Directed Graphs Without Rainbow Triangles

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2025-02-04 DOI:10.1002/jgt.23224

Sebastian Babiński, Andrzej Grzesik, Magdalena Prorok

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引用次数: 0

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 this variant we consider $c$ graphs on a common vertex set, think of each graph as edges in a distinct color, and want to determine the smallest number of edges in each color which guarantees the 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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没有彩虹三角形的有向图

图论中最基本的结果之一是曼特尔定理，它决定了n阶无三角形图的最大边数。最近，这个问题的一个不同版本得到了解决。在这种变体中，我们考虑一个公共顶点集上的c个图，将每个图视为具有不同颜色的边，并想要确定每种颜色中保证彩虹三角形存在的最小边数。在这里，我们解决了无彩虹三角形的有向图的类似问题，无论彩虹三角形是有向的还是可传递的，对于任意数量的颜色。c = 3和c≥4的构造和证明本质上是不同的禁止三角形的类型。此外，我们还解决了有向图设置中的类似问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .