## 3 Edge Decomposition of Bipartite Circulant Graphs \(C_{2 nm,nm}\) Based on Cartesian Products
Hence, from (13) and (14), we can conclude that  
\[\left\{(b_{i}-a_{i})(d_{j}-c_{j}):i\in\mathbb{Z}_{4},j\in\mathbb{Z}_{4m}\right\} =\mathbb{Z}_{4}\times\mathbb{Z}_{4m}. \tag{15}\]  
Then, let \(\varphi:\mathbb{Z}_{4}\times\mathbb{Z}_{4m}\rightarrow\mathbb{Z}_{16m}\) be a one-one mapping defined by  
\[\varphi\big{(}a_{i}c_{j}\big{)}=e_{4 mi+j}=4 ma_{i}+c_{j},i\in\mathbb{Z}_{4},j \in\mathbb{Z}_{4m}, \tag{16}\]  
and \(\psi:\mathbb{Z}_{4}\times\mathbb{Z}_{4m}\rightarrow\mathbb{Z}_{16m}\) be a one-one mapping defined by  
\[\psi\big{(}b_{i}d_{j}\big{)}=f_{4 mi+j}=4 mb_{i}+d_{j},i\in\mathbb{Z}_{4},j\in \mathbb{Z}_{4m}. \tag{17}\]  
Hence, we can obtain the following chain \((e_{0},f_{0}),(e_{1},f_{1}),\ldots,(e_{16m-1},f_{16m-1})\). The translation for the previous chain is \((e_{0}+\sigma,f_{0}+\sigma),(e_{1}+\sigma,f_{1}+\sigma),\ldots,\)\((e_{16m-1}+\sigma,f_{16m-1}+\sigma),\sigma\in\mathbb{Z}_{16m}\), where the additions are calculated modulo \(16m\). Let \(\pi:\mathbb{Z}_{16m}\rightarrow\mathbb{Z}_{32m}\) be a one-one mapping defined by  
\[\pi(f_{a,\sigma})=16m+f_{a,\sigma},f_{a,\sigma}=f_{a}+\sigma,\alpha,\sigma\in \mathbb{Z}_{16m}. \tag{18}\]  
We now find novel chains of ordered pairs that are defined by \((e_{0}+\sigma,\pi(f_{0,\sigma}))\), \((e_{1}+\sigma,\pi(f_{1,\sigma})),\ldots,(e_{16m-1}+\sigma,\pi(f_{16m-1}, \sigma))\). These novel chains represent the edge decomposition of \(C_{32m,16m}\), i.e., \(E(\mathbb{C}_{\sigma})=\{(e_{0}+\sigma,\pi(f_{0,\sigma})),(e_{1}+\sigma,\pi(f_{1,\sigma})),\ldots,\)\((e_{16m-1}+\sigma,\pi(f_{16m-1},\sigma))\},\sigma\in\mathbb{Z}_{16m}\) and \(\cup_{\sigma\in\mathbb{Z}_{16m}}E(\mathbb{C}_{\sigma})=E(C_{32m,16m})\) that can be proved as follows. Let \(\lambda_{i}=f_{i}-e_{i}\), which is unique for every pair \((e_{i},f_{i});i\in\mathbb{Z}_{16m}\). For \(u\
eq v\in\mathbb{Z}_{16m}\), let \((e_{i}+u,f_{i}+u)\in E(\mathbb{C}_{u})\) and \((e_{i}+v,f_{i}+v)\in E(\mathbb{C}_{v})\). Suppose that \(|E(\mathbb{C}_{u})\cap\ E(\mathbb{C}_{v})|\
eq 0\) such that there is at least one edge \((e_{i},f_{i})\in\{E(\mathbb{C}_{u})\cap\ E(\mathbb{C}_{v})\}\). 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 \(16m\) to the second component of the ordered pairs \((e_{0}+\sigma,f_{0}+\sigma),(e_{1}+\sigma,f_{1}+\sigma),\ldots,(e_{16m-1}+\sigma,f _{16m-1}+\sigma),\sigma\in\mathbb{Z}_{16m}\) constructs the edge decomposition of \(C_{32m,16m}\). Hence, \(\cup_{\sigma\in\mathbb{Z}_{16m}}E(\mathbb{C}_{\sigma})=E(C_{32m,16m})\). \(\square\)  
**Lemma 2**. : _There is an edge decomposition of \(C_{16,8}\) by the graph \(2K_{2,2}\) and an edge decomposition of \(C_{8m,4m}\) by the graph \(mK_{2,2}\), then there is an edge decomposition of \(C_{64m,32m}\) by the graph \(\mathbb{C}\cong 2 mK_{4,4};gcd(m,3)=1\)._  
Proof. : The edge set of \(2K_{2,2}\) can be represented by \((a_{0},b_{0}+8),(a_{1},b_{1}+8),(a_{2},b_{2}+8)\), \((a_{3},b_{3}+8),(a_{4},b_{4}+8),(a_{5},b_{5}+8),(a_{6},b_{6}+8),(a_{7},b_{7}+8)\), where  
\[a_{i}= \begin{cases}0\;if\;i=0,4\\ 1\;if\;i=1,5\\ 2\;if\;i=2,6^{\prime}\\ 3\;if\;i=3,7\end{cases}  \tag{19}\] \[b_{i}=a_{i}+i.\]  
Hence, \(\{b_{i}-a_{i}:i\in\mathbb{Z}_{8}\}=\mathbb{Z}_{8}\). The edge set of the second graph \(mK_{2,2}\) can be represented by \((c_{0},\;d_{0}+4m),(c_{1},\;d_{1}+4m)\)\(,\ldots,\;(c_{4m-1},\;d_{4m-1}+4m)\), where  
\[c_{i}= \begin{cases}i\;\;\;\;\;\;\;\;\;if\;i<2m\\ 2m+i\;if\;i\geq 2m\end{cases}  \tag{20}\] \[d_{i}=c_{i}+i.\]  
Hence, \(\{d_{i}-c_{i}:i\in\mathbb{Z}_{4m}\}=\mathbb{Z}_{4m}\).