## 3 Edge Decomposition of Bipartite Circulant Graphs \(C_{2 nm,nm}\) Based on Cartesian Products
Let \(\pi:\mathbb{Z}_{4 nm}\rightarrow\mathbb{Z}_{8 nm}\); it is a one-one mapping defined by  
\[\pi(f_{\alpha,\sigma})=4 nm+f_{\alpha,\sigma},f_{\alpha,\sigma}=f_{\alpha}+ \sigma,\alpha,\sigma\in\mathbb{Z}_{4 nm}. \tag{12}\]  
We now find novel chains of ordered pairs that are defined by \((e_{0}+\sigma,\pi(f_{0,\sigma}))\), \((e_{1}+\sigma,\pi(f_{1,\sigma})),\ldots,(e_{4 nm-1}+\sigma,\pi(f_{4 nm-1,\sigma }))\). These novel chains represent the edge decomposition of \(C_{8 nm,Anm}\) i.e., \(E(\mathbb{C}_{\sigma})=\{(e_{0}+\sigma,\pi(f_{0,\sigma})),(e_{1}+\sigma,\pi( f_{1,\sigma})),\ldots,\)\((e_{4 nm-1}+\sigma,\pi(f_{4 nm-1,\sigma}))\},\sigma\in\mathbb{Z}_{4 nm}\) and \(\cup_{\sigma\in\mathbb{Z}_{4 nm}}E(\mathbb{C}_{\sigma})=E(C_{8 nm,Anm})\) that can be proved as follows. Let \(\lambda_{i}=f_{i}-e_{i}\), which is unique for every pair \((e_{i},f_{i});i\in\mathbb{Z}_{4 nm}\). For \(u\
eq v\in\mathbb{Z}_{4 nm}\), 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 \(4 nm\) to the second component of the ordered pairs \((e_{0}+\sigma,f_{0}+\sigma)\), \((e_{1}+\sigma,f_{1}+\sigma),\ldots,(e_{4 nm-1}+\sigma,f_{4 nm-1}+\sigma),\sigma\in \mathbb{Z}_{4 nm}\) constructs the edge decomposition of \(C_{8 nm,Anm}\). Hence, \(\cup_{\sigma\in\mathbb{Z}_{4 nm}}E(\mathbb{C}_{\sigma})=E(C_{8 nm,Anm})\). \(\square\)  
**Lemma 1**. : _There is an edge decomposition of \(C_{8,4}\) by the graph \(K_{2,2}\) and an edge decomposition of \(C_{8m,4m}\) by the graph \(mK_{2,2}\), then there is an edge decomposition of \(C_{32m,16m}\) by the graph \(\mathbb{C}\cong mK_{4,4};gcd(m,3)=1\)._  
Proof. : The edge set of \(K_{2,2}\) can be represented by \((a_{0},b_{0}+4),(a_{1},b_{1}+4),(a_{2},b_{2}+4)\), \((a_{3},b_{3}+4)\), where  
\[ \begin{array}{c}a_{i}=\begin{cases}0\ if\ i=0,2\\ 1\ if\ i=1,3^{\prime}\\ b_{i}=a_{i}+i.\end{cases}\\ \end{array}  \tag{13}\]  
Hence, \(\{b_{i}-a_{i}:i\in\mathbb{Z}_{4}\}=\mathbb{Z}_{4}\). 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  
\[ \begin{array}{c}c_{i}=\begin{cases}i&if\ i<2m\\ 2m+i&if\ i\geq 2m^{\prime},\\ d_{i}=c_{i}+i.\end{cases}\\ \end{array}  \tag{14}\]  
Hence, \(\{d_{i}-c_{i}:i\in\mathbb{Z}_{4m}\}=\mathbb{Z}_{4m}\). Suppose that the first graph \(K_{2,2}\) is represented by the vector \(v(K_{2,2})=(0,1,0,1)\), and the second graph \(mK_{2,2}\) is represented by the vector \(w(mK_{2,2})=(c_{0},c_{1},c_{2},\ldots,c_{4m-1})\), then \(v(K_{2,2})\times w(mK_{2,2})=(a_{0}c_{0},a_{0}c_{1},\ldots,a_{p}c_{q},\ldots,a _{3}c_{4m-1})\), where \(p\in\mathbb{Z}_{4}\) and \(q\in\mathbb{Z}_{4m}\). Then, construct the ordered pairs \((a_{0}c_{0},b_{0}d_{0})\), \((a_{0}c_{1},b_{0}d_{1}),\ldots,\left(a_{p}c_{q},p_{0}d_{q}\right)\), \(\ldots,(a_{3}c_{4m-1},b_{3}d_{4m-1})\) where \(p\in\mathbb{Z}_{4}\) and \(q\in\mathbb{Z}_{4m}\).