## 3 Edge Decomposition of Bipartite Circulant Graphs \(C_{2 nm,nm}\) Based on Cartesian Products  
In this section, if graph \(G\) is represented by the vector \(v(G)=(a_{0},a_{1},a_{2},\ldots,a_{n-1})\) and graph \(H\) is represented by the vector \(w(H)=(c_{0},c_{1},c_{2},\ldots,c_{n-1})\), then \(v(G)\times w(H)=(a_{0}c_{0},a_{0}c_{1},\ldots,a_{p}c_{q},\ldots,a_{n-1}c_{n-1})\), where \(p\in\mathbb{Z}_{n}\) and \(q\in\mathbb{Z}_{m}\). The edge set of graph \(\mathbb{C}\) can be constructed based on \(\big{(}a_{0}c_{0},a_{0}c_{1},\ldots,a_{p}c_{q},\ldots,a_{n-1}c_{n-1}\big{)}\), as shown in the following proposition. Proposition 1: _If there is an edge decomposition of \(C_{2n,n}\) by the graph \(G\) and an edge decomposition of \(C_{2m,m}\) by the graph \(H\), then there is an edge decomposition of \(C_{2 nm,nm}\) by the graph \(\mathbb{C}\)._  
Proof: Let \(i,a_{i},b_{i}\in\mathbb{Z}_{n}=\{0,1,\ldots,n-1\}\). Make the following an ordered pair chain: \((a_{0},b_{0}),(a_{1},b_{1}),\ldots,(a_{n-1},b_{n-1})\), where the following condition  
\[\{b_{i}-a_{i}:i\in\mathbb{Z}_{n}\}=\mathbb{Z}_{n} \tag{1}\]  
is satisfied, and the difference between \(b_{i}\) and \(a_{i}\) is calculated modulo \(n\). Let \(j,c_{j},d_{j}\in\mathbb{Z}_{m}=\{0,1,\ldots,m-1\}\). Make the following an ordered pair chain: \((c_{0},d_{0}),(c_{1},d_{1}),\ldots,(c_{m-1},d_{m-1})\), where the following condition  
\[\{d_{j}-c_{j}:j\in\mathbb{Z}_{m}\}=\mathbb{Z}_{m} \tag{2}\]  
is satisfied, and the difference between \(d_{j}\) and \(c_{j}\) is calculated modulo \(m\). The edge set of the first graph \(G\) can be represented by \((a_{0},b_{0}+n),(a_{1},b_{1}+n),\ldots,(a_{n-1},b_{n-1}+n)\) and the edge set of the second graph \(H\) can be represented by \((c_{0},d_{0}+m),(c_{1},d_{1}+m),\ldots,(c_{m-1},d_{m-1}+m)\). Suppose that the first graph \(G\) is represented by the vector \(v(G)=(a_{0},a_{1},a_{2},\ldots,a_{n-1})\), and the second graph \(H\) is represented by the vector \(w(H)=(c_{0},c_{1},c_{2},\ldots,c_{m-1})\), then \(v(G)\times w(H)=\big{(}a_{0}c_{0},\ a_{0}c_{1},\ldots,\ a_{p}c_{q},\ldots,\ a_{n-1}c_{m-1}\big{)}\), where \(p\in\mathbb{Z}_{n}\) and \(q\in\mathbb{Z}_{m}\). Then, construct the ordered pairs \((a_{0}c_{0},b_{0}d_{0}),(a_{0}c_{1},b_{0}d_{1}),\ldots,\big{(}a_{p}c_{q},b_{p} d_{q}),\ldots,(a_{n-1}c_{m-1},b_{n-1}d_{m-1})\), where \(p\in\mathbb{Z}_{n}\) and \(q\in\mathbb{Z}_{m}\). Hence, from (1) and (2), we can conclude that  
\[\{(b_{i}-a_{i})\big{(}d_{j}-c_{j}\big{)}:i\in\mathbb{Z}_{n},j\in\mathbb{Z}_{m} \}=\mathbb{Z}_{n}\times\mathbb{Z}_{m}. \tag{3}\]  
Then, let \(\varphi:\mathbb{Z}_{n}\times\mathbb{Z}_{m}\rightarrow\mathbb{Z}_{nm}\) be a one-one mapping defined by  
\[\varphi\big{(}a_{i}c_{j}\big{)}=e_{mi+j}=ma_{i}+c_{j},i\in\mathbb{Z}_{m},j\in \mathbb{Z}_{m}, \tag{4}\]  
and \(\psi:\mathbb{Z}_{n}\times\mathbb{Z}_{m}\rightarrow\mathbb{Z}_{nm}\) be a one-one mapping defined by  
\[\psi\big{(}b_{i}d_{j}\big{)}=f_{mi+j}=mb_{i}+d_{j},i\in\mathbb{Z}_{n},j\in \mathbb{Z}_{m}. \tag{5}\]  
Hence, we can obtain the following chain \((e_{0},f_{0}),(e_{1},f_{1}),\ldots,(e_{nm-1},f_{nm-1})\). The translation for the previous chain is \((e_{0}+\sigma,f_{0}+\sigma),(e_{1}+\sigma,f_{1}+\sigma),\ldots,(e_{nm-1}+ \sigma,f_{nm-1}+\sigma)\), \(\sigma\in\mathbb{Z}_{nm}\), where the additions are calculated modulo \(nm\). Let \(\pi:\mathbb{Z}_{nm}\rightarrow\mathbb{Z}_{2 nm}\); it is a one-one mapping defined by  
\[\pi(f_{\alpha,\sigma})=nm+f_{\alpha,\sigma},f_{\alpha,\sigma}=f_{\alpha}+ \sigma,\alpha,\sigma\in\mathbb{Z}_{nm}. \tag{6}\]  
We now find novel chains of ordered pairs that are defined by \((e_{0}+\sigma,\pi(f_{0,\sigma})),\)\((e_{1}+\sigma,\pi(f_{1,\sigma})),\ldots,(e_{nm-1}+\sigma,\pi(f_{nm-1,\sigma}))\).