# The Locating-Chromatic Number of Origami Graphs
## 2 Results and Discussions
### Locating-Chromatic Number for Subdivision Outer Edge of Origami Graphs
\begin{cases}0,&\text{for the second component, odd }i,3\leq i\leq n,n\geq 7\\ &\text{for the third component, even }i,2\leq i\leq\left\lceil\frac{n}{2}\right\rceil-2,n \geq 7\\ &\text{for the third component, even }i,\left\lceil\frac{n}{2}\right\rceil+2\leq i\leq n-1,n \geq 7\\ &\text{for the fourth component, }i=1\\ &\text{for the fifth component, }i=\left\lceil\frac{n}{2}\right\rceil\\ 2,&\text{for the third component, }i=\left\lceil\frac{n}{2}\right\rceil\\ i-1,&\text{for the fourth component, }2\leq i\leq\left\lceil\frac{n}{2}\right\rceil-1,n \geq 7\\ n-i+1,&\text{for the fourth component, odd }i,\left\lceil\frac{n}{2}\right\rceil+1\leq i\leq n,n \geq 7\\ \left\lceil\frac{n}{2}\right\rceil-1,&\text{for the fourth component, }i=\left\lceil\frac{n}{2}\right\rceil\\ \left\lceil\frac{n}{2}\right\rceil-i,&\text{for the fifth component, }1\leq i\leq\left\lceil\frac{n}{2}\right\rceil-1,n \geq 7\\ i-\left\lceil\frac{n}{2}\right\rceil,&\text{for the fifth component, }\left\lceil\frac{n}{2}\right\rceil+1\leq i\leq n,n \geq 7\\ 1,&\text{otherwise }.\end{cases} \]  
\[c_{\Pi}(v_{i})= \begin{cases}0,&\text{for the second component, even }i,2\leq i\leq n-1,n\geq 7\\ &\text{for the third component, odd }i,1\leq i\leq n,n\geq 7\\ 2,&\text{for the first component, }1\leq i\leq n,n\geq 7\\ i,&\text{for the fourth component, }1\leq i\leq\left\lceil\frac{n}{2}\right\rceil,n \geq 7\\ n-i+2&\text{for the fourth component, }\left\lceil\frac{n}{2}\right\rceil+1\leq i\leq n,n \geq 7\\ \left\lceil\frac{n}{2}\right\rceil-i+1&\text{for the fifth component, }1\leq i\leq\left\lceil\frac{n}{2}\right\rceil,n \geq 7\\ i-\left\lceil\frac{n}{2}\right\rceil+1&\text{for the fifth component, }\left\lceil\frac{n}{2}\right\rceil+1\leq i \leq n,n\geq 7\\ 1,&\text{otherwise }.\end{cases} \]  
\[c_{\Pi}(w_{i})= \begin{cases}0,&\text{for the first component, }1\leq i\leq n,n\geq 7\\ 2,&\text{for the third component, }i=\left\lceil\frac{n}{2}\right\rceil-1\text{ and }i=n\\ i,&\text{for the fourth component, }1\leq i\leq\left\lceil\frac{n}{2}\right\rceil,n \geq 7\\ n-i+1,&\text{for the fourth component, }\left\lceil\frac{n}{2}\right\rceil+1\leq i \leq n,n\geq 7\\ \left\lceil\frac{n}{2}\right\rceil-i,&\text{for the fifth component, }1\leq i\leq\left\lceil\frac{n}{2}\right\rceil-1,n \geq 7\\ i-\left\lceil\frac{n}{2}\right\rceil+1,&\text{otherwise }.\end{cases} \]  
\[c_{\Pi}(x_{i})= \begin{cases}0,&\text{for the second component, odd }i,1\leq i\leq n,n\geq 7\\ &\text{for the third component, even }i,2\leq i\leq n-1,n\geq 7\\ i+1,&\text{for the fourth component, }1\leq i\leq\left\lceil\frac{n}{2}\right\rceil-1,n \geq 7\\ n-i+2,&\text{for the fourth component, }\left\lceil\frac{n}{2}\right\rceil\leq i \leq n,n\geq 7\\ \left\lceil\frac{n}{2}\right\rceil-i+2,&\text{for the fifth component, }1\leq i\leq\left\lceil\frac{n}{2}\right\rceil,n \geq 7\\ i-\left\lceil\frac{n}{2}\right\rceil+2,&\text{for the fifth component, }\left\lceil\frac{n}{2}\right\rceil+1\leq i \leq n,n\geq 7\\ 1,&\text{otherwise }.\end{cases} \]  
Since for \(n\) odd all vertices have different color codes, \(c\) is a locating coloring for a subdivision of the outer edge of origami graph \(O_{n}^{*}\), so that \(\chi_{L}(O_{n}^{*})\leq 5\), for \(\left\lceil\frac{n}{2}\right\rceil\) even, \(n\geq 7\)  
**Subcase c.** (even \(n\)), for \(\frac{n}{2}\) odd, \(n\geq 6\). Let \(c\) be a coloring for a subdivision outer edge of origami graph \(O_{n}^{*}\), for \(\frac{n}{2}\) odd, and \(n\geq 6\); we make the partition \(\Pi\) of \(V(O_{n}^{*})\) :  
\(C_{1}=\{w_{i}|1\leq i\leq\frac{n}{2}-1\}\cup\{w_{i}|\frac{n}{2}+1\leq i\leq n\}\);  
\(C_{2}=\{u_{i}|\text{ for odd }i,3\leq i\leq n-1\}\cup\{v_{i}|\text{ for even }i,2\leq i\leq n\}\cup\{x_{i}|\text{ for odd }i,1\leq i\leq n-1\}\);  
\(C_{3}=\{u_{i}|\text{ for even }i,2\leq i\leq n\}\cup\{v_{i}|\text{ for odd }i,1\leq i\leq n-1\}\cup\{x_{i}|\text{ for even }i,2\leq i\leq n\}\}\);  
\(C_{4}=\{u_{1}\}\);  
\(C_{5}=\{w_{\frac{n}{2}}\}\). For \(\frac{n}{2}\) odd, the color codes of all the vertices of \(V(O_{n}^{*})\) are:  
\[c_{\Pi}(u_{i})=