# On Orthogonal Double Covers and Decompositions of Complete Bipartite Graphs by Caterpillar Graphs
## 3 Main Results  
In Theorem 1, we construct and prove the existence of the symmetric starter of an \(ODC\) of \(K_{xy,xy}\) by \(\mathbb{C}_{2}^{1,y-2}(S_{x})\) using a general symmetric vector. For more information, see Figure 4.  
**Theorem 1**. : _For all integers \(y\geq 3\) and \(x\geq 2\), then the vector \(v(\mathbb{C}_{2}^{1,y-2}(S_{x}))=(0,0,\ldots,0,x,x,x,\ldots,x)\in\mathbb{Z}_{yx}^ {yx}\), is a symmetric starter vector of an ODC of \(K_{xy,xy}\) by \(\mathbb{C}_{2}^{1,y-2}(S_{x})\)._  
**Proof. ** From the vector \(v\Big{(}\mathbb{C}_{2}^{1,y-2}(S_{x})\Big{)}\), we can define:  
\[v_{i}= \begin{cases}0&\text{if}\quad i<x,\text{or}\\ x&\text{otherwise}.\end{cases}  \tag{1}\]  
Hence,  
\[v_{-i}= \begin{cases}x&\text{if}\quad 1\leq i\leq x(y-1),\text{or}\\ 0&\text{otherwise}.\end{cases}  \tag{2}\]  
Then,  
\[v_{i}-v_{-i}+i=\left\{ \begin{array}{lll}0&\text{if}\quad i=0,\text{or}\\ i-x&\text{if}\quad 1\leq i<x,\text{or}\\ i&\text{if}\quad x\leq i\leq x(y-1),\text{or}\\ i+x&\text{otherwise}.\end{array} \right. \tag{3}\]  
Note that \(\left\{v_{i}-v_{-i}+i:i\in\mathbb{Z}_{yx}\right\}=\mathbb{Z}_{yx}\). \(\square\)  
Example 2 is a direct application of Theorem 1. **Example 2**. : _Let \(y=4\) and \(x=3\); so, the vector \(v\Big{(}\mathbb{C}_{2}^{1,2}(S_{3})\Big{)}=(0,0,0,3,3,3,3,3,3,3,3)\). Then, there exists an ODC of \(K_{12,12}\) by \(\mathbb{C}_{2}^{1,2}(S_{3})\). The edge set of \(v\Big{(}\mathbb{C}_{2}^{1,2}(S_{3})\Big{)}\) is defined by \(E(\mathbb{C}_{2}^{1,2}(S_{3}))=\{((v_{i})_{0},(v_{i}+i)_{1}):i\in\mathbb{Z}_{ 12}\}=\{(0_{0},0_{1}),(0_{0},1_{1}),(0_{0},2_{1}),(3_{0},6_{1}),(3_{0},7_{1}), (3_{0},8_{1}),(3_{0},9_{1}),(3_{0},10_{1}),(3_{0},11_{1}),(3_{0},1_{1}),(3_{0}, 1_{1}),(3_{0},2_{1})\}\), see Figure 5._  
In Theorem 2, we construct the symmetric starter of an \(ODC\) of \(K_{2 xy,2 xy}\) by \(2\mathbb{C}_{2}^{1,y-2}(S_{x})\) using a general symmetric vector. There are two cases depending on the value of \(x\). For more information, see Figures 6 and 7. **Theorem 2**. : _For all integers \(y\geq 3\) and \(x\geq 2\), then the vector \(v(2\mathbb{C}_{2}^{1,y-2}(S_{x}))=(x,x-2,x,\allowbreak x-2,\allowbreak\ldots,x,x-2,3x,3x-2,3x,3x-2,\allowbreak\ldots,3x,3x-2,\allowbreak\ldots,3x,3x-2)\in \mathbb{Z}_{2 xy}^{2 xy}\), is a symmetric starter vector of an ODC of \(K_{2 xy,2 xy}\) by \(2\mathbb{C}_{2}^{1,y-2}(S_{x})\)._  
**Proof. ** From the vector \(v\left(2C_{2}^{1,y-2}(S_{x})\right)\), we can define:  
\[v_{i}=\left\{ \begin{array}{lll}x&\text{if}&i=0,2,4,\ldots,2x-2,\text{or}\\ x-2&\text{if}&i=1,3,5,\ldots,2x-1,\text{or}\\ 3x&\text{if}&i=2x,2x+2,2x+4,\ldots,2 yx-2,\text{or}\\ 3x-2&\text{if}&i=2x+1,2x+3,2x+5,\ldots,2 yx-1.\end{array} \right. \tag{4}\]  
Hence,  
\[v_{-i}=\left\{ \begin{array}{lll}3x-2&\text{if}&i=1,3,5,\ldots,2 yx-2x-1,\text {or}\\ 3x&\text{if}&i=2,4,6,\ldots,2 yx-2x,\text{or}\\ x-2&\text{if}&i=2 yx-2x+1,2 yx-2x+3,\ldots,2 yx-1,\text{or}\\ x&\text{if}&i=2 yx-2x+2,2 yx-2x+4,\ldots,2 yx.\end{array} \right. \tag{5}\]  
Then,  
\[v_{i}-v_{-i}+i=\left\{ \begin{array}{lll}0&\text{if}&i=0,\text{or}\\ i-2x&\text{if}&i=1,2,3,\ldots,2x-1,\text{or}\\ i&\text{if}&i=2x,2x+1,2x+2,\ldots,2x(y-1),\text{or}\\ i+2x&\text{otherwise}.\end{array} \right. \tag{6}\]  
Note that \(\{v_{i}-v_{-i}+i:\,i\in\mathbb{Z}_{2 yx}\}=\mathbb{Z}_{2 yx}\). \(\square\)  
Example 3 is a direct application of Theorem 2. **Example 3. **_Let \(y=3\) and \(x=3\); so, the vector \(v(2C_{2}^{1,1}(S_{3}))=(3,1,3,1,3,1,9,7,9,7,9,7,\allowbreak 9,7,9,7,\allowbreak 9,7, 9,7)\). Then, there exists an \(ODC\) of \(K_{18,18}\) by \(2C_{2}^{1,1}(S_{3})\); for more information, see Figure 8._  
**Theorem 3**. : _Let \(y\geq 3\) and \(x\geq 2\) be integers, then, the vector \(v(\mathbb{C}_{2}^{1,y-2}(K_{2,x}))=(0,1,0,1,\ldots,0,\)\(1,2x,2x+1,2x,2x+1,\ldots,2x,2x+1)\in\mathbb{Z}_{2 xy}^{2 xy}\), is a symmetric starter vector of an ODC of \(K_{2 xy,2 xy}\) by \(\mathbb{C}_{2}^{1,y-2}(K_{2,x})\)._  
Proof. : From the vector \(v\left(\mathbb{C}_{2}^{1,y-2}(K_{2,x})\right)\), we can define:  
\[v_{i}=\left\{