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 $$n\ge 5$$ .