Directed Path 3-Arc-Connectivity of Cartesian Product Digraphs

Abstract

Let D=(V(D),A(D)) be a digraph of order n and let r∈S⊆V(D) with 2≤|S|≤n. A directed (S,r)-Steiner path (or an (S,r)-path for short) is a directed path P beginning at r such that S⊆V(P). Arc-disjoint between two (S,r)-paths is characterized by the absence of common arcs. Let λS,rp(D) be the maximum number of arc-disjoint (S,r)-paths in D. The directed path k-arc-connectivity of D is defined as λkp(D)=min{λS,rp(D)∣S⊆V(D),S=k,r∈S}. In this paper, we shall investigate the directed path 3-arc-connectivity of Cartesian product λ3p(G□H) and prove that if G and H are two digraphs such that δ0(G)≥4, δ0(H)≥4, and κ(G)≥2, κ(H)≥2, then λ3p(G□H)≥min2κ(G),2κ(H); moreover, this bound is sharp. We also obtain exact values for λ3p(G□H) for some digraph classes G and H, and most of these digraphs are symmetric.