Rainbow Triangles in Arc-Colored Tournaments

Abstract

Let $T_{n}$ be an arc-colored tournament of order n. The maximum monochromatic indegree $\varDelta ^{-mon}(T_{n})$ (resp. outdegree $\varDelta ^{+mon}(T_{n})$) 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 $\varDelta ^{-mon}(T_n)\le \varDelta ^{+mon}(T_n)$ is contained in at least $\frac{\delta (v)(n-\delta (v)-i(T_n))}{2}-[\varDelta ^{-mon}(T_{n})(n-1)+\varDelta ^{+mon}(T_{n})d^+(v)]$ 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.