Rainbow Antistrong Connection in Tournaments

Abstract

An arc-coloured digraph is rainbow antistrong connected if any two distinct vertices u, v are connected by both a forward antidirected (u, v)-trail and a forward antidirected (v, u)-trail which do not use two arcs with the same colour. The rainbow antistrong connection number of a digraph D is the minimum number of colours needed to make the digraph rainbow antistrong connected, denoted by $\overset{\rightarrow }{rac}(D)$. An arc-coloured digraph is strong rainbow antistrong connected if any two distinct vertices u, v are connected by both a forward antidirected (u, v)-geodesic trail and a forward antidirected (v, u)-geodesic trail which do not use two arcs with the same colour. The strong rainbow antistrong connection number of a digraph D, denoted by $\overset{\rightarrow }{srac}(D)$, is the minimum number of colours needed to make the digraph strong rainbow antistrong connected. In this paper, we prove that for any antistrong tournament $T_n$ with n vertices $\overset{\rightarrow }{rac}(T_n)\ge 3$ and $\overset{\rightarrow }{srac}(T_n)\ge 3$, and we construct tournaments $T_n$ with $\overset{\rightarrow }{rac}(T_n)=\overset{\rightarrow }{srac}(T_n)=3$ for every $n\ge 18$. Then, we prove that for any antistrong tournament $T_n$ whose diameter is at least 4, $\overset{\rightarrow }{rac}(T_n)\le 7$, and we construct tournaments $T_n$ whose diameter is 3 with $\overset{\rightarrow }{rac}(T_n)=7$ for every $n\ge 5$.