# Directed Path 3-Arc-Connectivity of Cartesian Product Digraphs
## 3 A General Lower Bound
In this case, our overall goal is that we will use arc-disjoint paths between \(x\) and \(y\) in \(G(v_{1})\), \(y\) and its out-neighbors in \(H(u_{2})\), \(z\) and its in-neighbors in \(G(v_{2})\), and combine them together to form the required arc-disjoint paths. The general idea of the proof process is briefly described in Figure 2.0 The vertices and paths contained in Figure 3 are explained below. Considering \(S_{1}=\{x,y\}\), \(r_{1}=x\), it is known that there exist at least \(\kappa(G)\) internally disjoint \((S_{1},r_{1})\)-paths in \(G(v_{1})\), denoted as \(\widetilde{P}_{1i}\) (\(i\in[\kappa(G)]\)). Let \(S_{2}=\{y,u_{2,2}\}\), \(r_{2}=y\), and there exist at least \(\kappa(G)\) internally disjoint \((S_{2},r_{2})\)-paths in \(H(u_{2})\), denoted as \(\widetilde{P}_{1i}\) (\(i\in[\kappa(G)]\)). For each \(i\in[\kappa(G)]\), let \(u_{2,f_{i}}\) be the out-neighbor of \(y\) in \(\widetilde{P}_{1i}\); clearly these out-neighbors are distinct. For each \(i\in[\kappa(G)]\), an out-neighbor \(u_{b,f_{i}}\) of \(u_{2,f_{i}}\) in \(G(v_{f_{i}})\) can be chosen, with \(b\
eq 1\). In \(H(u_{b})\), with \(S_{3}=\{u_{b,1},u_{b,2}\}\) and \(r_{3}=u_{b,1}\). \(\widetilde{P}_{1i}\) is the \((S_{3},r_{3})\)-path corresponding to \(\widetilde{P}_{1i}\). In \(\widetilde{P}_{1i}^{\prime}\), the path from vertex \(u_{b,f_{i}}\) to \(u_{b,2}\) is denoted \(\widetilde{P}_{1i}^{\prime\prime}\). With \(S_{4}=\{u_{b,2},z\}\) and \(r_{4}=u_{b,2}\), it is known that there exist at least \(\kappa(G)\) internally disjoint \((S_{4},r_{4})\)-paths in \(G(v_{2})\), denoted as \(\widetilde{P}_{1i}\) (\(i\in[\kappa(G)]\)). If \(u_{2,f_{k}}=u_{2,2}\), then \(u_{2,2}\
otin\widetilde{P}_{1k}\). The arc-disjoint \((S,r)\)-paths can be constructed as  
\(P_{1i}=\widetilde{P}_{1i}\cup\widetilde{P}_{1i}^{\prime\prime}\cup\widetilde{ P}_{1i}\cup\{yu_{2,f_{i}},u_{2,f_{i}}u_{b,f_{i}}\}\), \(i\in[\kappa(G)]\). Likewise, we can identify \(\kappa(G)\) arc-disjoint \((S,r)\)-paths from \(x\) to \(z\) and subsequently to \(y\). Consequently, we can derive \(2\kappa(G)\) arc-disjoint \((S,r)\)-paths. **Case 3. **: Let \(x\), \(y\) and \(z\) be in different \(H(u_{i})\) and \(G(v_{j})\) for some \(i\in[n]\), \(j\in[m]\). Without loss of generality, we can assume that \(x=u_{1,1}\), \(y=u_{2,2}\), \(z=u_{3,3}\). In this case, our overall goal is that, we will use arc-disjoint paths between \(x\) and its out-neighbors in \(H(u_{1})\), \(y\) and its out-neighbors in \(H(u_{2})\), \(z\) and its in-neighbors in \(G(v_{3})\), \(x\) and its out-neighbors in \(G(v_{1})\), \(y\) and its out-neighbors in \(G(v_{2})\), \(z\) and its in-neighbors in \(H(u_{3})\), and combine them together to form the required arc-disjoint paths. The general idea of the proof process is briefly described in Figure 2.1 The vertices and paths contained in Figure 4 are explained below. Considering \(S_{1}=\{x,u_{2,1}\}\), \(r_{1}=x\), it is known that there exist at least \(\kappa(G)\) internally disjoint \((S_{1},r_{1})\)-paths in \(G(v_{1})\), denoted as \(\widetilde{P}_{1i}\) (\(i\in[\kappa(G)]\)). Let \(S_{2}=\{y,u_{2,2}\}\), \(r_{2}=y\), and there exist at least \(\kappa(G)\) internally disjoint \((S_{2},r_{2})\)-paths in \(H(u  
and there exist at least \(\kappa(G)\) internally disjoint \((S_{2},r_{2})\)-paths in \(H(u_{2})\), denoted as \(\widehat{P}_{1i}\) (\(i\in[\kappa(G)]\)). Considering \(S^{\prime}_{1}=\{x,u_{1,2}\}\), \(r^{\prime}_{1}=x\), it is known that there exist at least \(\kappa(G)\) internally disjoint \((S^{\prime}_{1},r^{\prime}_{1})\)-paths in \(H(u_{1})\), denoted as \(\widehat{P}_{2j}\) (\(j\in[\kappa(G)]\)). Let \(S^{\prime}_{2}=\{u_{1,2},y\}\), \(r^{\prime}_{2}=u_{1,2}\), and there exist at least \(\kappa(G)\) internally disjoint \((S^{\prime}_{2},r^{\prime}_{2})\)-paths in \(G(v_{2})\), denoted as \(\widehat{P}_{2j}\) (\(j\in[\kappa(G)]\)).