Strong subgraph 2-arc-connectivity and arc-strong connectivity of Cartesian product of digraphs

Let \(D=(V,A)\) be a digraph of order \(n\), \(S\) a subset of \(V\) of size \(k\) and \(2\le k\leq n\). A strong subgraph \(H\) of \(D\) is called an \(S\)-strong subgraph if \(S\subseteq V(H)\). A pair of \(S\)-strong subgraphs \(D_1\) and \(D_2\) are said to be arc-disjoint if \(A(D_1)\cap A(D_2)=\emptyset\). Let \(\lambda_S(D)\) be the maximum number of arc-disjoint \(S\)-strong subgraphs in \(D\). The strong subgraph \(k\)-arc-connectivity is defined as $$\lambda_k(D)=\min\{\lambda_S(D)\mid S\subseteq V(D), |S|=k\}.$$ The parameter \(\lambda_k(D)\) 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 \(\lambda(G\Box H)\) 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 \(\lambda_2(G\Box H)\) and prove that \( \min\left \{ \lambda \left ( G \right ) \left | H \right | , \lambda \left ( H \right ) \left |G \right |,\delta ^{+ } \left ( G \right )+ \delta ^{+ } \left ( H \right ),\delta ^{- } \left ( G \right )+ \delta ^{- } \left ( H \right ) \right \}\ge\lambda_2(G\Box H)\ge \lambda_2(G)+\lambda_2(H)-1.\) The upper bound for \(\lambda_2(G\Box H)\) is sharp and is a simple corollary of the formula for \(\lambda(G\Box H)\). The lower bound for \(\lambda_2(G\Box H)\) is either sharp or almost sharp i.e. differs by 1 from the sharp bound. We also obtain exact values for \(\lambda_2(G\Box H)\), where \(G\) and \(H\) are digraphs from some digraph families.