Strong subgraph 2-arc-connectivity and arc-strong connectivity of Cartesian product of digraphs
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Let D=(V,A) be a digraph of order n, S a subset of V of size k and 2≤ k≤ n. A strong subgraph H of D is called an S-strong subgraph if S⊆ V(H). A pair of S-strong subgraphs D_1 and D_2 are said to be arc-disjoint if A(D_1)∩ A(D_2)=∅. Let λ_S(D) be the maximum number of arc-disjoint S-strong subgraphs in D. The strong subgraph k-arc-connectivity is defined as λ_k(D)=min{λ_S(D)| S⊆ V(D), |S|=k}. The parameter λ_k(D) can be seen as a generalization of classical edge-connectivity of undirected graphs. In this paper, we first obtain a formula for the arc-connectivity of Cartesian product λ(G H) of two digraphs G and H generalizing a formula for edge-connectivity of Cartesian product of two undirected graphs obtained by Xu and Yang (2006). Then we study the strong subgraph 2-arc-connectivity of Cartesian product λ_2(G H) and prove that min{λ ( G ) | H | , λ ( H ) |G |,δ ^+ ( G )+ δ ^+ ( H ),δ ^- ( G )+ δ ^- ( H ) }≥λ_2(G H)≥λ_2(G)+λ_2(H)-1. The upper bound for λ_2(G H) is sharp and is a simple corollary of the formula for λ(G H). The lower bound for λ_2(G H) is either sharp or almost sharp i.e. differs by 1 from the sharp bound. We also obtain exact values for λ_2(G H), where G and H are digraphs from some digraph families.
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