## 3 Edge Decomposition of Bipartite Circulant Graphs \(C_{2 nm,nm}\) Based on Cartesian Products
Suppose that the first graph \(K_{2,2}\) is represented by the vector \(v(2K_{2,2})=(0,1,2,3,0,1,2,3)\), and the second graph \(mK_{2,2}\) is represented by the vector \(w(mK_{2,2})=(c_{0},c_{1},\ldots,c_{q},\ldots,c_{4m-1})\), then \(v(2K_{2,2})\times w(mK_{2,2})=\left(a_{0}c_{0},a_{0}c_{1},\ldots,a_{p}c_{q}, \ldots,a_{7}c_{4m-1}\right)\), where \(p\in\mathbb{Z}_{8}\) and \(q\in\mathbb{Z}_{4m}\). Then, construct the ordered pairs \((a_{0}c_{0},b_{0}d_{0}),(a_{0}c_{1},b_{0}d_{1}),\ldots,\left(a_{p}c_{q},b_{p}d _{q}\right),\ldots,(a_{7}c_{4m-1},\;b_{7}d_{4m-1})\) where \(p\in\mathbb{Z}_{8}\) and \(q\in\mathbb{Z}_{4m}\). Hence, from (19) and (20), we can conclude that  
\[\left\{(b_{i}-a_{i})\left(d_{j}-c_{j}\right):i\in\mathbb{Z}_{8},j\in\mathbb{Z} _{4m}\right\}=\mathbb{Z}_{8}\times\mathbb{Z}_{4m}. \tag{21}\]  
Then, let \(\varphi:\mathbb{Z}_{8}\times\mathbb{Z}_{4m}\rightarrow\mathbb{Z}_{32m}\) be a one-one mapping defined by  
\[\varphi\left(a_{i}c_{j}\right)=e_{4 mi+j}=4 ma_{i}+c_{j},i\in\mathbb{Z}_{8},j\in \mathbb{Z}_{4m}, \tag{22}\]  
and \(\psi:\mathbb{Z}_{8}\times\mathbb{Z}_{4m}\rightarrow\mathbb{Z}_{32m}\) be a one-one mapping defined by  
\[\psi\left(b_{i}d_{j}\right)=f_{4 mi+j}=4 mb_{i}+d_{j},i\in\mathbb{Z}_{8},j\in \mathbb{Z}_{4m}. \tag{23}\]  
Hence, we can obtain the following chain \((e_{0},f_{0}),(e_{1},f_{1}),\ldots,(e_{32m-1},f_{32m-1})\). The translation for the previous chain is \((e_{0}+\sigma,f_{0}+\sigma),(e_{1}+\sigma,f_{1}+\sigma),\ldots,\)\((e_{32m-1}+\sigma,f_{32m-1}+\sigma),\sigma\in\mathbb{Z}_{32m}\), where the additions are calculated modulo \(32m\). Let \(\pi:\mathbb{Z}_{32m}\rightarrow\mathbb{Z}_{64m}\); it is a one-one mapping defined by  
\[\pi(f_{a,\sigma})=32m+f_{a,\sigma},f_{a,\sigma}=f_{a}+\sigma,\alpha,\sigma\in \mathbb{Z}_{32m}. \tag{24}\]  
We now find novel chains of ordered pairs that are defined by \((e_{0}+\sigma,\pi(f_{0,\sigma}))\), \((e_{1}+\sigma,\pi(f_{1,\sigma})),\ldots,(e_{32m-1}+\sigma,\pi(f_{32m-1,\sigma}))\). These novel chains represent the edge decomposition of \(C_{64m,32m}\) i.e., \(E(\mathbb{C}_{\sigma})=\{(e_{0}+\sigma,\pi(f_{0,\sigma})),(e_{1}+\sigma,\pi(f_ {1,\sigma})),\ldots,\)\((e_{32m-1}+\sigma,\pi(f_{32m-1,\sigma}))\},\sigma\in\mathbb{Z}_{32m}\) and \(\cup_{\sigma\in\mathbb{Z}_{32m}}E(\mathbb{C}_{\sigma})=E(C_{64m,32m})\) that can be proved as follows. Let \(\lambda_{i}=f_{i}-e_{i}\), which is unique for every pair \((e_{i},f_{i});i\in\mathbb{Z}_{32m}\). For \(u\
eq v\in\mathbb{Z}_{32m}\), 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 orderedpairs \((\epsilon_{0}+\sigma,f_{0}+\sigma)\), \((e_{1}+\sigma,f_{1}+\sigma),\ldots,(e_{32m-1}+\sigma,f_{32m-1}+\sigma),\sigma\in \mathbb{Z}_{32m}\) constructs the edge decomposition of \(C_{64m,32m}\). Hence, \(\cup_{\sigma\in\mathbb{Z}_{22m}}E(\mathbb{C}_{\sigma})=E(C_{64m,32m})\). \(\square\)  
Theorem 2.2: _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 \(K_{1,2}\cup\ K_{1,2(m-1)}\), then there is an edge decomposition of \(C_{4 nm,2 mn}\) by the graph \(K_{1,2n}\cup\ K_{1,2n(m-1)};m\geq 1,n\geq 2\)._  
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  
\[a_{i}=0;i\in\mathbb{Z}_{n}, \tag{25}\] \[b_{i}=a_{i}+i.\]  
Hence, \(\{b_{i}-a_{i}:i\in\mathbb{Z}_{n}\}=\mathbb{Z}_{n}\).