Rainbow transitive triangles in arc-colored digraphs

Abstract

A subdigraph of an arc-colored digraph is rainbow if its all arcs have distinct colors. For two digraphs $D$ and $H$, let $rb(D,H)$ be the minimum integer such that every arc-colored digraph $D^C$ with $c\left(D\right)\ge rb(D,H)$ contains a rainbow copy of $H$, where $c\left(D\right)$ is the number of colors of $D^C$. Let $\overleftrightarrow{K_n}$ be the digraph obtained from the complete graph $K_n$ by replacing each edge $uv$ with a pair of symmetric arcs $(u,v)$ and $(v,u)$, and let $\overrightarrow{T_3}$ be the transitive triangle. In this paper we determine $rb(\overleftrightarrow{K_n},\overrightarrow{T_3})$ and characterize the corresponding extremal arc-colorings of $\overleftrightarrow{K_n}$. Further, we prove that an arc-colored digraph $D^C$ on $n$ vertices contains a rainbow $\overrightarrow{T_3}$ if $a\left(D\right)+c\left(D\right)\ge a\left(\overleftrightarrow{K_n}\right)+rb(\overleftrightarrow{K_n},\overrightarrow{T_3})$. Moreover, if $a\left(D\right)+c\left(D\right)=a\left(\overleftrightarrow{K_n}\right)+rb(\overleftrightarrow{K_n},\overrightarrow{T_3})-1$ and $D^C$ contains no rainbow $\overrightarrow{T_3}$’s, then $D\cong \overleftrightarrow{K_n}$.