Rainbow triangles in arc-colored tournaments

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