# Directed Path 3-Arc-Connectivity of Cartesian Product Digraphs
## 3 A General Lower Bound
: Let \(S=\{x,y,z\}\), \(r=x\), and we only examine the case where \(x\), \(y\), and \(z\) are not all within the same \(\overleftrightarrow{T}_{n}(u_{i})\) or the same \(\overleftrightarrow{K}_{m}(v_{j})\) 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 \(x=u_{1,1}\), \(y=u_{2,2}\), \(z=u_{3,3}\). We can obtain \(m\) arc-disjoint \((S,r)\)-paths in \(\overleftrightarrow{T}_{n}\square\overleftrightarrow{K}_{m}\), say \(P_{1}\), \(P_{2}\), \(\ldots\), \(P_{i}\) (\(4<i\leq m\)), \(P_{m-1}\), \(P_{m}\) such that  
\(P_{1}:xu_{2,1}yu_{3,2}z\), \(P_{2}:xu_{1,2}yu_{2,3}z\), \(P_{3}:xu_{1,3}u_{2,3}yu_{2,1}u_{3,1}z\),  
\(P_{4}:xu_{1,4}u_{2,4}u_{3,4}zu_{3,2}y\), \(P_{i}:xu_{1,i}u_{2,i}u_{3,i}zu_{3,i-1}u_{2,i-1}y\). Then we have \(m=\min\{\delta^{+}(D),\delta^{-}(D)\}\geq\lambda_{3}^{p}(\overrightarrow{T}_{ n}\square\overleftrightarrow{K}_{m})\geq m\). This completes the proof. Proposition 5.: _We have \(\lambda_{3}^{p}(\overrightarrow{C}_{n}\square\overleftrightarrow{C}_{m})=2\)._  
Proof. : Let \(S=\{x,y,z\}\), \(r=x\), and we only examine the case where \(x\), \(y\), and \(z\) are not all within the same \(\overrightarrow{C}_{n}(u_{i})\) or the same \(\overrightarrow{C}_{m}(v_{j})\) 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 \(x=u_{1,1}\), \(y=u_{2,2}\), \(z=u_{3,3}\). We can obtain two arc-disjoint \((S,r)\)-paths in \(\overrightarrow{C}_{n}\square\overleftrightarrow{C}_{m}\), say \(P_{1}\) and \(P_{2}\) such that  
\(P_{1}:xu_{2,1}yu_{3,2}z\), \(P_{2}:xu_{1,2}yu_{2,3}z\). Then we have \(2=\min\{\delta^{+}(D),\delta^{-}(D)\}\geq\lambda_{3}^{p}(\overrightarrow{C}_{ n}\square\overleftrightarrow{C}_{m})\geq 2\). This completes the proof. Proposition 6.: _We have \(\lambda_{3}^{p}(\overrightarrow{C}_{n}\square\overleftrightarrow{C}_{m})=3\), with \(m\geq 3\)._  
Proof. : Let \(S=\{x,y,z\}\), \(r=x\), and we only examine the case where \(x\), \(y\), and \(z\) are not all within the same \(\overrightarrow{C}_{n}(u_{i})\) or the same \(\overleftrightarrow{C}_{m}(v_{j})\) 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 \(x=u_{1,1}\), \(y=u_{2,2}\), \(z=u_{3,3}\). We can obtain three arc-disjoint \((S,r)\)-paths in \(\overrightarrow{C}_{n}\square\overleftrightarrow{C}_{m}\), say \(P_{1}\), \(P_{2}\), \(P_{3}\) such that  
\(P_{1}:xu_{2,1}yu_{3,2}z\), \(P_{2}:xu_{1,2}yu_{2,3}z\),  
\(P_{3}:xu_{m,1}u_{m,2}u_{m-1,2}\ldots u_{3,2}yu_{1,2}u_{1,3}u_{m,3}u_{m-1,3} \ldots z\). Then we have \(3=\min\{\delta^{+}(D),\delta^{-}(D)\}\geq\lambda_{3}^{p}(\overrightarrow{C}_{ n}\square\overleftrightarrow{C}_{m})\geq 3\). This completes the proof. Proposition 7.: _We have \(\lambda_{3}^{p}(\overleftrightarrow{C}_{n}\square\overleftrightarrow{C}_{m})=4\), with \(n\geq 3\), \(m\geq 3\)._Proof. : Let \(S=\{x,y,z\}\), \(r=x\), and we only examine the case where \(x,y,\) and \(z\) are not all within the same \(\overrightarrow{C}_{n}(u_{i})\) or the same \(\overleftrightarrow{C}_{m}(v_{j})\) 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 \(x=u_{1,1},y=u_{2,2},z=u_{3,3}\). We can obtain four arc-disjoint \((S,r)\)-paths in \(\overleftrightarrow{C}_{n}\square\overleftrightarrow{C}_{m}\), say \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\) such that  
\(P_{1}:xu_{2,1}yu_{3,2}z\), \(P_{2}:xu_{1,2}yu_{2,3}z\),  
\(P_{3}:xu_{m,1}u_{m,2}u_{m,3}u_{m-1,3}\ldots zu_{3,2}y\), \(P_{4}:xu_{1,2}u_{2,n}u_{3,n}zu_{2,3}y\). Then we have \(\mathbf{4}=\min\{\delta^{+}(D),\delta^{-}(D)\}\geq\lambda_{3}^{p}( \overleftrightarrow{C}_{n}\square\overleftrightarrow{C}_{m})\geq 4\). This completes the proof. Proposition 8: _We have \(\lambda_{3}^{p}(\overrightarrow{C}_{n}\square\overleftrightarrow{T}_{m})=2\)._  
Proof.