# Directed Path 3-Arc-Connectivity of Cartesian Product Digraphs
## 3 A General Lower Bound
In \(H(u_{k_{j}})\), let \(S^{\prime}_{6_{j}}=\{u_{k_{j},2},u_{k_{j},d}\}\), \(r^{\prime}_{6_{j}}=u_{k_{j},2}\), and at least \(\kappa(G)\) internally disjoint \((S^{\prime}_{6_{j}},r^{\prime}_{6_{j}})\)-paths are known to exist. Then, one of the paths \(\tilde{P}_{2j}\) (\(j\in[\kappa(G)]\)) is chosen, where \(u_{k_{j},1}\
otin\tilde{P}_{2j}\). If \(u_{s_{j},2}=u_{k_{j},2}\) (\(t\in[\kappa(G)]\)), \(P_{2t}=\widetilde{P}_{2t}\cup\widetilde{P}_{2t}\cup\{zu_{k_{j},2}u_{s_{j},2}y\}\). And if \(u_{k_{j},2}=y\) (\(l\in[\kappa(G)]\)), \(P_{2t}=\widetilde{P}_{2t}\cup\widetilde{P}_{2t}\cup\{zy\}\). If \(u_{s_{g},d}u_{s_{g},2}\
otin\widetilde{P}_{1i}\) in the path \(\widetilde{P}_{1i}\). Let  
\(P_{1i}=\widetilde{P}_{1i}\cup\overline{P}_{2i}\cup\widetilde{P}_{1i}\cup \widetilde{P}_{1i}\),  
\(P_{2j}=\widetilde{P}_{2j}\cup\widetilde{P}_{2j}\cup\widetilde{P}_{2j}\cup \widetilde{P}_{2j}\cup\{zu_{k_{j},2},u_{s_{j},2}u_{s_{j},2}y\}\). If an arc \(u_{s_{g},d}u_{s_{g},2}\) is in path \(\widetilde{P}_{1i}\), an in-neighbor \(u_{s_{g},f}\) of \(u_{s_{g},2}\) can be found in \(H(u_{3})\) such that \(u_{s_{g},f}u_{s_{g},2}\
otin\widetilde{P}_{1i}\) and \(f\in[m]\setminus\{c,1\}\). In \(G(\tilde{r}_{g})\), let \(\overline{P}^{\prime\prime}_{2g}\) be the \((S^{\prime}_{7},r^{\prime}_{7})\)-path corresponding to \(\overline{P}^{\prime\prime}_{2g}\), where \(S^{\prime}_{7}=\{u_{k_{g},f},u_{s_{g},f}\}\), \(r^{\prime}_{7}=u_{k_{g},f}\). In \(H(u_{k_{g}})\), let \(S^{\prime}_{8}=\{u_{k_{g},2},u_{k_{g},f}\}\), \(r^{\prime}_{8}=u_{k_{g},2}\), and at least \(\kappa(G)\) internally disjoint \((S^{\prime}_{7},r^{\prime}_{7})\)-paths are known to exist. Then, one of the paths \(\tilde{P}^{\prime}_{2g}\) is chosen, and let \(u_{k_{j},1}\
otin\tilde{P}^{\prime}_{2g}\). Let  
\(P_{2g}=\widetilde{P}_{2g}\cup\overline{P}^{\prime\prime\prime}_{2g}\cup \widetilde{P}_{2g}\cup\tilde{P}^{\prime}_{2g}\cup\{zu_{k_{g},2},u_{s_{g},f}u_{ s_{g},2}u_{s_{g},2}y\}\). Hence, we obtain \(2\kappa(G)\) arc-disjoint \((S,r)\)-paths. Now we prove that this bound is sharp. By Proposition 1, \(\lambda^{p}_{3}(\overleftrightarrow{K}_{n}\square\overleftrightarrow{K}_{m})= n+m-2\). By Lemma 2, \(\kappa(\overleftrightarrow{K}_{n})=n-1\). So we have \(\lambda^{p}_{3}(\overleftrightarrow{K}_{n}\square\overleftrightarrow{K}_{n}) =2\kappa(\overleftrightarrow{K}_{n})=2n-2\), with \(n\geq 5\). Therefore, the lower bound holds and is sharp. **4. Exact Values for Digraph Classes**  
In this section, we aim to determine precise values for the directed path 3-arc-connectivity of the Cartesian product of two digraphs within specific digraph classes. **Proposition 1**. : _We have \(\lambda^{p}_{3}(\overleftrightarrow{K}_{n}\square\overleftrightarrow{K}_{m}) =n+m-2\)._  
Proof. : Consider \(S=\{x,y,z\}\) and \(r=x\). We will focus solely on scenarios where \(x\), \(y\), and \(z\) do not all belong to the same \(\overleftrightarrow{K}_{m}(u_{i})\) or the same \(\overleftrightarrow{K}_{n}(v_{j})\) for any \(i\in[n],j\in[m]\). The rationale for the remaining cases follows a similar line of argument. Without loss of generality, let us assume \(x=u_{1,1},y=u_{2,2}\), \(z=u_{3,3}\). It is feasible to derive \(n+m-2\) arc-disjoint \((S,r)\)-paths in \(\overleftrightarrow{K}_{n}\square\overleftrightarrow{K}_{m}\), say \(P_{1}\),\(P_{2}\),..., \(P_{a}\) (\(a=\min\{i+1,3<i\leq n\}\)), \(P_{i+1}\) (\(4<i\leq n\)),..., \(P_{b}\) (\(b=\min\{n+j-2,3<j\leq n\}\)), \(P_{n+j-2}\) (\(4<j\leq m\)) (as shown in Figure 8) such that  
\(P_{1}:xu_{2,1}yu_{3,2}\), \(P_{2}:xu_{1,2}yu_{2,3}z\), \(P_{3}:xu_{1,3}zu_{3,2}y\),  
\(P_{4}:xu_{3,1}zu_{2,3}y\), \(P_{a}:xu_{4,1}u_{3,2}u_{1,4}u_{3,1}u_{1,2}u_{4,2}y\),  
\(P_{i+1}:xu_{i,1}u_{i,3}zu_{i-1,3}u_{i-1,2}u_{i,2}y\), \(P_{n+j-2}:xu_{1,j}u_{3,j}zu_{3,j-1}u_{2,j-1}u_{2,j}y\). Now, we add two cases to prove that the proposition holds, so as to show that the proposition has no constraint conditions. First, let \(n=m=4\). We can assume that \(x=u_{1,1}\), \(y=u_{2,2}\), \(z=u_{3,3}\).