On optimal orientations of cartesian products of graphs (ii): complete graphs and even cycles Original Research Article

For a graph G, let D(G) be the family of strong orientations of G. Define d⇀(G)=min{d(D) /D∈D(G)} and ρ(G)=d⇀(G)−d(G), where d(D) (resp., d(G)) denotes the diameter of the digraph D (resp., graph G). Let G×H denote the cartesian product of the graphs G and H, Kp the complete graph of order p and Cp the cycle of order p. In this paper, we show that ρ(K2×C2m)=2, ρ(Kn×C2m)=1 for n=3,4,5,7, and ρ(Kn×C2m)=0 for most cases otherwise.