## 3 Edge Decomposition of Bipartite Circulant Graphs \(C_{2 nm,nm}\) Based on Cartesian Products
From the translation definition, we have \((e_{i}+u-u,f_{i}+u-u)=(e_{i},f_{i}),(e_{i}+v-v,f_{i}+v-v)=(e_{i},f_{i});\lambda _{i}=f_{i}-e_{i}\) is unique for every edge in \(\mathbb{C}_{0}\). Hence, \(|E(\mathbb{C}_{u})\cap\ E(\mathbb{C}_{v})|\
eq 0\). Then, the addition of \(8 nm\) to the second component of the ordered pairs \((e_{0}+\sigma,f_{0}+\sigma),(e_{1}+\sigma,f_{1}+\sigma),\)\(\ldots,(e_{8 nm-1}+\sigma,f_{8 nm-1}+\sigma),\sigma\in\mathbb{Z}_{8 nm}\) constructs the edge decomposition of \(C_{16 nm,8 nm}\). Hence, \(\cup_{\sigma\in\mathbb{Z}_{8 nm}}E(\mathbb{C}_{\sigma})=E(C_{16 nm,8 nm})\). \(\square\)  
**Theorem 5**. : _There is an edge decomposition of \(C_{2n,n}\) by the graph \(K_{1,n}\) and an edge decomposition of \(C_{4m,2m}\) by the graph \(K_{1,2}\cup\ K_{2,m-1}\), then there is an edge decomposition of \(C_{4 nm,2 nm}\) by the graph \(K_{2,n}\cup\ K_{2,n(m-1)};m,n\geq 2\)._  
Proof. : The edge set of \(K_{1,n}\) can be represented by \((a_{0},b_{0}+n),(a_{1},b_{1}+n),\ldots,\)\((a_{n-1},b_{n-1}+n)\), where  
\[ \begin{array}{c}a_{i}=0;i\in\mathbb{Z}_{n},\\ b_{i}=a_{i}+i.\end{array}  \tag{43}\]  
Hence, \(\{b_{i}-a_{i}:i\in\mathbb{Z}_{n}\}=\mathbb{Z}_{n}\). The edge set of the second graph \(K_{1,2}\cup\ K_{2,m-1}\) can be represented by \((c_{0},d_{0}+2m)\), \((c_{1},d_{1}+2m),\ldots,(c_{2m-1},d_{2m-1}+2m)\), where  
\[c_{i}=\left\{ \begin{array}{ccc}0&if&i=0;\\ 1&if&1\leq i\leq m-1;\\ m&if&i=m;\\ m+1&if&m+1\leq i\leq 2m-1\end{array} \right.\]  
\[d_{i}=c_{i}+i. \tag{44}\]  
Hence, \(\{d_{j}-c_{j}:j\in\mathbb{Z}_{2m}\}=\mathbb{Z}_{2m}\). Suppose that the first graph \(K_{1,n}\) is represented by the vector \(v(K_{1,n})\)\(=(a_{0},a_{1},a_{2},\ldots,a_{n-1})\), and the second graph \(mK_{2,2}\) is represented by the vector \(v(K_{1,2}\cup\ K_{2,m-1})=(c_{0},c_{1},c_{2},\ldots,c_{2m-1})\), then \(v(K_{1,n})\times w(K_{1,2}\cup\ K_{2,m-1})=(a_{0}c_{0},a_{0}c_{1},\)\(\ldots,a_{p}c_{q},\ldots a_{n-1}c_{2m-1})\), where \(p\in\mathbb{Z}_{m}\) and \(q\in\mathbb{Z}_{2m}\). Then, construct the ordered pairs \((a_{0}c_{0},b_{0}d_{0}),(a_{0}c_{1},b_{0}d_{1}),\ldots,(a_{p}c_{q},b_{p}d_{q}), \ldots,(a_{n-1}c_{2m-1},b_{n-1}d_{2m-1})\), where \(p\in\mathbb{Z}_{n}\) and \(q\in\mathbb{Z}_{2m}\). Hence, from (43) and (44), we can conclude that  
\[\left\{(b_{i}-a_{i})\big{(}d_{j}-c_{j}\big{)}:i\in\mathbb{Z}_{n},j\in\mathbb{Z }_{2m}\right\}=\mathbb{Z}_{n}\times\mathbb{Z}_{2m}. \tag{45}\]  
Then, let \(\varphi:\mathbb{Z}_{n}\times\mathbb{Z}_{2m}\rightarrow\mathbb{Z}_{2 nm}\) be a one-one mapping defined by  
\[\varphi\big{(}a_{i}c_{j}\big{)}=e_{2 mi+j}=2 ma_{i}+c_{j},i\in\mathbb{Z}_{n},j \in\mathbb{Z}_{2m}, \tag{46}\]  
and \(\psi:\mathbb{Z}_{n}\times\mathbb{Z}_{2m}\rightarrow\mathbb{Z}_{2 nm}\) be a one-one mapping defined by  
\[\psi\big{(}b_{i}d_{j}\big{)}=f_{2 mi+j}=2 mb_{i}+d_{j},i\in\mathbb{Z}_{n},j\in \mathbb{Z}_{2m}. \tag{47}\]  
Hence, we can obtain the following chain \((e_{0},f_{0}),(e_{1},f_{1}),\ldots,(e_{2 nm-1},f_{2 nm-1})\). The translation for the previous chain is \((e_{0}+\sigma,f_{0}+\sigma),(e_{1}+\sigma,f_{1}+\sigma),\ldots,\)\((e_{2 nm-1}+\sigma,f_{2 nm-1}+\sigma),\sigma\in\mathbb{Z}_{2 mn}\), where the additions are calculated modulo nm. Let \(\pi:\mathbb{Z}_{2 nm}\rightarrow\mathbb{Z}_{4 nm}\) be a one-one mapping defined by  
\[\pi(f_{\alpha,\sigma})=2 nm+f_{\alpha,\sigma},f_{\alpha,\sigma}=f_{\alpha}+\sigma, \alpha,\sigma\in\mathbb{Z}_{2 nm}. \tag{48}\]We now find novel chains of ordered pairs that are defined by \((e_{0}+\sigma,\pi(f_{0,\sigma}))\), \((e_{1}+\sigma,\pi(f_{1,\sigma})),\ldots,(e_{2 nm-1}+\sigma,\pi(f_{2 nm-1,\sigma}))\). These novel chains represent the edge decomposition of \(C_{4 nm,2 mn}\) i.e., \(E(\mathbb{C}_{\sigma})=\{(e_{0}+\sigma,\pi(f_{0,\sigma})),(e_{1}+\sigma,\pi(f_ {1,\sigma})),\ldots,\)\((e_{2 nm-1}+\sigma,\pi(f_{2 nm-1,\sigma}))\},\sigma\in\mathbb{Z}_{2 nm}\) and \(\cup_{\sigma\in\mathbb{Z}_{2 nm}}E(\mathbb{C}_{\sigma})=E(C_{4 nm,2 nm})\) that can be proved as follows.