Rainbow triangles in arc-colored tournaments

TL;DR

This paper investigates conditions under which arc-colored tournaments contain rainbow triangles, providing bounds based on monochromatic degrees and irregularity, and demonstrating the optimality of some conditions.

Contribution

It introduces new degree-based criteria ensuring the existence of rainbow triangles in arc-colored tournaments, extending previous results in the field.

Findings

Each vertex is contained in a number of rainbow triangles depending on degree and irregularity.

Maximum monochromatic degree conditions guarantee rainbow triangles passing through specific vertices.

Examples show some conditions are optimal and cannot be improved.

Abstract

Let $T_{n}$ be an arc-colored tournament of order $n$. The maximum monochromatic indegree $\Delta^{-mon}(T_{n})$ (resp. outdegree $\Delta^{+mon}(T_{n})$) of $T_{n}$ is the maximum number of in-arcs (resp. out-arcs) of a same color incident to a vertex of $T_{n}$. The irregularity $i(T_{n})$ of $T_{n}$ is the maximum difference between the indegree and outdegree of a vertex of $T_{n}$. A subdigraph $H$ of an arc-colored digraph $D$ is called rainbow if each pair of arcs in $H$ have distinct colors. In this paper, we show that each vertex $v$ in an arc-colored tournament $T_{n}$ with $\Delta^{-mon}(T_n)\leq\Delta^{+mon}(T_n)$ is contained in at least $\frac{\delta(v)(n-\delta(v)-i(T_n))}{2}-[\Delta^{-mon}(T_{n})(n-1)+\Delta^{+mon}(T_{n})d^+(v)]$ rainbow triangles, where $\delta(v)=\min\{d^+(v), d^-(v)\}$. We also give some maximum monochromatic degree conditions for $T_{n}$ to contain rainbow triangles, and to contain rainbow triangles passing through a given vertex. Finally, we present some examples showing that some of the conditions in our results are best possible. Keywords: arc-colored tournament, rainbow triangle, maximum monochromatic indegree (outdegree), irregularity