# Directed Path 3-Arc-Connectivity of Cartesian Product Digraphs
## 3 A General Lower Bound
In \(G(v_{1})\), with \(S_{1}^{\prime}=\{x,u_{1,1}\}\), and \(r_{1}^{\prime}=x\), it is known that there exist at least \(\kappa(G)\) internally disjoint \((S_{1}^{\prime},r_{1}^{\prime})\)-paths, denoted as \(\widehat{P}_{2j}\). In \(H(u_{1})\), with \(S_{2}^{\prime}=\{u_{1,1},u_{1,2}\}\), and \(r_{2}^{\prime}=u_{1,1}\), it is known that there exist at least \(\kappa(G)\) internally disjoint \((S_{2}^{\prime},r_{2}^{\prime})\)-paths, denoted as \(\widetilde{P}_{2j}\). In \(G(v_{2})\), with \(S_{3}^{\prime}=\{u_{1,2},y\}\), and \(r_{3}^{\prime}=u_{1,2}\), it is known that there exist at least \(\kappa(G)\) internally disjoint \((S_{3}^{\prime},r_{3}^{\prime})\)-paths, denoted as \(\widetilde{P}_{2j}\). Let \(u_{s_{i},2}\), \(u_{s_{i},c}\), \(u_{2,f_{i}}\), \(u_{b,f_{i}}\), \(\widetilde{P}_{1i}\), \(\widetilde{P}_{2j}\), \(\overline{P}_{1i}^{\prime\prime}\) and \(\overline{P}_{2j}^{\prime\prime}\) be the same as in Case 3. If \(u_{s_{k},2}=u_{3,2}\), then \(u_{3,2}\
otin\widetilde{P}_{1k}\)\((k\in[\kappa(G)])\). If \(u_{2,f_{r}}=u_{2,3}\), then \(u_{2,3}\
otin\widetilde{P}_{2r}\)\((r\in[\kappa(G)])\). If \(u_{1,3}\in\widetilde{P}_{2h}\), then \(u_{1,3}\
otin\widetilde{P}_{2h}\)\((h\in[\kappa(G)])\). If \(u_{b,1}\in\widetilde{P}_{2l}^{\prime\prime}\), then \(u_{b,1}\
otin\widetilde{P}_{2l}\)\((t\in[\kappa(G)])\). If \(u_{1,3}\in\widetilde{P}_{2l}\), then \(u_{1,3}\
otin\widetilde{P}_{2l}\)\((l\in[\kappa(G)])\). **Subcase 4.1**. : If there exists no vertex \(u_{2,f_{j}}=x\). Let  
\(P_{1i}=\widetilde{P}_{1i}\cup\overline{P}_{1i}^{\prime\prime}\cup\widetilde{P }_{1i}\cup\{yu_{s_{i},2},u_{s_{i},2},u_{s_{i},c}\}\),  
\(P_{2j}=\widetilde{P}_{2j}\cup\overline{P}_{2j}^{\prime\prime}\cup\widehat{P }_{2j}\cup\widetilde{P}_{2j}\cup\{yu_{2,f_{j}},u_{2,f_{j}}u_{b,f_{j}}\}\). **Subcase 4.2**. : If there exists a vertex \(u_{2,f_{g}}=x\)\((g\in[\kappa(G)])\), then in \(G(v_{1})\), there exists an out-neighbor \(u_{b,1}\) of \(x\). If \(u_{b,1}\in\widehat{P}_{2j}\), this path is denoted by \(\widehat{P}_{2g}\). In \(H(u_{3})\), there exists an out-neighbor \(u_{3,g_{1}}\) of \(z\) such that \(g_{1}\in[m]\setminus\{c,2,1\}\). In \(G(v_{2})\), there exists an in-neighbor \(u_{g_{2},2}\) of \(y\) such that \(g_{2}\in[n]\setminus\{1,b,3\}\). If \(u_{g_{2},2}\in\widehat{P}_{2j}\), this path is denoted by \(\widetilde{P}_{2g}\). Then in \(H(u_{g_{2}})\), with \(S_{4}^{\prime}=\{u_{g_{2},g_{1}},u_{g_{2},2}\}\), and \(r_{4}^{\prime}=u_{g_{2},g_{1}}\), it is known that there are at least \(\kappa(G)\) internally disjoint \((S_{4}^{\prime},r_{4}^{\prime})\)-paths. One such \((S_{4}^{\prime},r_{4}^{\prime})\)-path is chosen, denoted as \(\widehat{P}_{g}\), with \(u_{g_{2},3}\
otin\widehat{P}_{g}\). In \(G(v_{g_{1}})\), with \(S_{5}^{\prime}=\{u_{3,g_{1}},u_{g_{2},g_{1}}\}\), and \(r_{5}^{\prime}=u_{3,g_{1}}\), it is known that there are at least \(\kappa(G)\) internally disjoint \((S_{5}^{\prime},r_{5}^{\prime})\)-paths. One such \((S_{5}^{\prime},r_{5}^{\prime})\)-path is chosen, denoted as \(\widetilde{P}_{g}\), with \(u_{b,g_{1}}\
otin\widetilde{P}_{g}\). Then, \(P_{2g}\) is constructed as  
\(P_{2g}=\overline{P}_{2g}^{\prime\prime}\cup\widetilde{P}_{2g}\cup\widetilde{P }_{g}\cup\widehat{P}_{g}\cup\{xu_{b,1},zu_{3,g_{1}},u_{g_{2},2}y\}\). The other paths are the same as Subcase 4.1. Then we obtain \(2\kappa(G)\) arc-disjoint \((S,r)\)-paths. Figure 2.4 Depiction of the arc-disjoint paths found in Case 4 of the proof of Theorem 1. **Case 5. **: Let \(x\) and \(y\) be in the same \(H(u_{i})\). Let \(y\) and \(z\) be in the same \(G(v_{j})\), for some \(i\in[n]\), \(j\in[m]\). Without loss of generality, we can assume that \(x=u_{1,1}\), \(y=u_{1,2}\), \(z=u_{2,2}\). In this case, our overall goal is that we will use arc-disjoint paths between \(x\) and \(y\) in \(H(u_{1})\), \(y\) and \(z\) in \(G(v_{2})\), \(x\) and its out-neighbors in \(G(v_{1})\), \(x\) and its out-neighbors in \(G(v_{1})\), \(z\) and \(y\) in \(G(v_{2})\), and combine them together to form the required arc-disjoint paths.