## 3 Edge Decomposition of Bipartite Circulant Graphs \(C_{2 nm,nm}\) Based on Cartesian Products
: _There is an edge decomposition of \(\mathbb{C}_{4n,2n}\) by the graph \(K_{1,2}\cup\ K_{1,2(n-1)}\) and an edge decomposition of \(\mathbb{C}_{8 nm,4 nm}\) by the graph \(mK_{2,2}\), then there is an edge decomposition of \(\mathbb{C}_{16 nm,8 nm}\) by the graph \(mK_{2,4}\cup\ mK_{2,4(n-1)};m,n\geq 2,gcd(m,3)=1\)._  
Proof. : The edge set of \(K_{1,2}\cup\ K_{1,2(n-1)}\) can be represented by \((a_{0},b_{0}+2n)\), \((a_{1},b_{1}+2n),\ldots,\)\((a_{2n-1},b_{2n-1}+2n)\), where  
\[a_{i}= \begin{cases}0&if\qquad\qquad\qquad i=0,n;\\ n&if\quad 1\leq i\leq n-1,n+1\leq i\leq 2n-1\end{cases} \]  
\[b_{i}=a_{i}+i. \tag{37}\]  
Hence, \(\{b_{i}-a_{i}:i\in\mathbb{Z}_{2n}\}=\mathbb{Z}_{2n}\). 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  
\[c_{i}= \begin{cases}i\qquad if\ i<2m\\ 2m+i\ if\ i\geq 2m.\\ d_{i}=c_{i}+i.\end{cases}  \tag{38}\]  
Hence, \(\{d_{i}-c_{i}:i\in\mathbb{Z}_{4m}\}=\mathbb{Z}_{4m}\). Suppose that the first graph \(K_{1,2}\cup K_{1,2(n-1)}\) is represented by the vector \(v\big{(}K_{1,2}\cup\ K_{1,2(n-1)}\big{)}=(a_{0},a_{1},a_{2},\ldots,a_{2n-1})\), and the second graph \(mK_{2,2}\) is represented by the vector \(v(mK_{2,2})=(c_{0},c_{1},c_{2},\ldots,c_{4m-1})\), then \(v\big{(}K_{1,2}\cup\ K_{1,2(n-1)}\big{)}\times w(mK_{2,2})=(a_{0}c_{0},a_{0}c_{ 1},\ldots,a_{p}c_{q},\ldots,a_{2n-1}c_{4m-1})\), where \(p\in\mathbb{Z}_{2n}\) and \(q\in\mathbb{Z}_{4m}\). Then, construct the ordered pairs \((a_{0}c_{0},b_{0}d_{0})\), \((a_{0}c_{1},b_{0}d_{1})\), \(\ldots,\big{(}a_{p}c_{q},b_{p}d_{q}\big{)}\), \(\ldots,(a_{2n-1}c_{4m-1},b_{2n-1}d_{4m-1})\), where \(p\in\mathbb{Z}_{2n}\) and \(q\in\mathbb{Z}_{4m}\). Hence, from (37) and (38), we can conclude that  
\[\big{\{}(b_{i}-a_{i})\big{(}d_{j}-c_{j}\big{)}:i\in\mathbb{Z}_{2n},j\in\mathbb{ Z}_{4m}\big{\}}=\mathbb{Z}_{2n}\times\mathbb{Z}_{4m}. \tag{39}\]  
Then, let \(\varphi:\mathbb{Z}_{2n}\times\mathbb{Z}_{4m}\rightarrow\mathbb{Z}_{8 nm}\) be a one-one mapping defined by  
\[\varphi\big{(}a_{i}c_{j}\big{)}=e_{4 mi+j}=4 ma_{i}+c_{j},i\in\mathbb{Z}_{2n},j \in\mathbb{Z}_{4m}, \tag{40}\]  
and \(\psi:\mathbb{Z}_{2n}\times\mathbb{Z}_{4m}\rightarrow\mathbb{Z}_{8 nm}\) be a one-one mapping defined by  
\[\psi\big{(}b_{i}d_{j}\big{)}=f_{4 mi+j}=4 mb_{i}+d_{j},i\in\mathbb{Z}_{2n},j\in \mathbb{Z}_{4m}. \tag{41}\]  
Hence, we can obtain the following chain \((e_{0},f_{0})\), \((e_{1},f_{1})\), \(\ldots,(e_{8 nm-1},f_{8 nm-1})\). The translation for the previous chain is \((e_{0}+\sigma,f_{0}+\sigma)\), \((e_{1}+\sigma,f_{1}+\sigma)\), \(\ldots\), \((e_{8 nm-1}+\sigma,f_{8 nm-1}+\sigma)\), \(\sigma\in\mathbb{Z}_{8 nm}\), where the additions are calculated modulo \(nm\). Let \(\pi:\mathbb{Z}_{8 nm}\rightarrow\mathbb{Z}_{16 nm}\); it is a one-one mapping defined by  
\[\pi(f_{a,\sigma})=8 nm+f_{a,\sigma},f_{a,\sigma}=f_{a}+\sigma,\alpha,\sigma\in \mathbb{Z}_{8 nm}. \tag{42}\]  
We now find novel chains of ordered pairs that are defined by \((e_{0}+\sigma,\pi(f_{0,\sigma}))\), \((e_{1}+\sigma,\pi(f_{1,\sigma}))\), \(\ldots,(e_{8 nm-1}+\sigma,\pi(f_{8 nm-1,\sigma}))\). These novel chains represent the edge de composition of \(C_{16 nm,8 nm}\) i.e., \(E(C_{\sigma})=\{(e_{0}+\sigma,\pi(f_{0,\sigma})),(e_{1}+\sigma,\pi(f_{1,\sigma}) ),\ldots,\)\((e_{8 nm-1}+\sigma,\pi(f_{8 nm-1,\sigma})),\sigma\in\mathbb{Z}_{8 nm}\) and \(\cup_{\sigma\in\mathbb{Z}_{8 nm}}E(C_{\sigma})=E(C_{16 nm,8 nm})\) that can be proved as follows. Let \(\lambda_{i}=f_{i}-e_{i}\), which is unique for every pair \((e_{i},f_{i});i\in\mathbb{Z}_{8 nm}\). For \(u\
eq v\in\mathbb{Z}_{8 nm}\), 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 \((e_{i},f_{i})\in\{E(\mathbb{C}_{u})\cap\ E(\mathbb{C}_{v})\}\).