# The Locating-Chromatic Number of Origami Graphs
## 2 Results and Discussions
### Locating-Chromatic Number for Subdivision Outer Edge of Origami Graphs  
**Theorem 2**. : _Let \(O_{n}^{*}\) be a subdivision outer edge of origami graphs for \(n\geq 3\). Then the locating-chromatic number of \(O_{n}^{*}\), \(\chi_{L}(O_{n}^{*})= \begin{cases}4,&\text{for }n=3\\ 5,&\text{otherwise}\end{cases} .\)_  
Proof. : Let \(O_{n}^{*}\), \(n\geq 3\) be a subdivision of an origami graph; \(O_{n}^{*}\) is a graph with \(V(O_{n}^{*})=\{u_{i},v_{i},x_{i},w_{i}:i\in\{1,...,n\}\}\) and \(E(O_{n}^{*})=\{u_{i}w_{i},u_{i}v_{i},v_{i}x_{i},x_{i}w_{i}:i\in\{1,...,n\}\}\cup \{u_{i}u_{i+1},w_{i}u_{i+1}:i\in\{1,...,n-1\}\}\cup\{u_{n}u_{1},w_{n}u_{1}\}\}\). Next, to prove the theorem, we consider the following two cases:  
**Case A. **\(\chi_{L}(O_{3}^{*})=4\)  
First, we determine the lower bound of \(\chi_{L}(O_{3}^{*})\). Without loss of generality, we assign \(A=\{c(u_{i}),c(v_{i}),c(x_{i}),c(w_{i}),c(u_{i+1})\}=\{1,2,3\}\). Then, there are three dominant vertices in \(A\), while we still have vertices on other \(A\) that must be colored. As a result, there will be two vertices with the same color codes. Thus, \(\chi_{L}(O_{3}^{*})\geq 4\). Next, we determine the upper bound of \(\chi_{L}(O_{3}^{*})\leq 4\). To show that 4 is an upper bound for the locating-chromatic number for a subdivision outer edge of origami graph \(O_{3}^{*}\), we describe a locating coloring \(c\) using four colors as follows:  
\(c(u_{i})=i,i=1,2,3\). \(c(v_{i})= \begin{cases}2,&\text{for }i=1,3\\ 3,&\text{for }i=2.\end{cases} \)  
\(c(w_{i})=4,i=1,2,3\). \(c(x_{i})=i,i=1,2,3\). The coloring \(c\) will create the partition \(\Pi\) on \(V(O_{3}^{*})\). We shall show that the color codes of all vertices in \(O_{3}^{*}\) are different. We have: \(c_{\Pi}(u_{1})=(0,1,1,1)\); \(c_{\Pi}(u_{2})=(1,0,1,1)\); \(c_{\Pi}(u_{3})=(1,1,0,1)\); \(c_{\Pi}(v_{1})=(1,0,2,2)\); \(c_{\Pi}(v_{2})=(2,1,0,2)\); \(c_{\Pi}(v_{3})=(2,0,1,2)\); \(c_{\Pi}(w_{1})=(1,1,2,0)\); \(c_{\Pi}(w_{2})=(2,1,1,0)\); \(c_{\Pi}(w_{3})=(1,2,1,0)\). \(c_{\Pi}(x_{1})=(0,1,3,1)\); \(c_{\Pi}(x_{2})=(3,0,1,1)\); \(c_{\Pi}(x_{3})=(2,1,0,1)\). Since the color codes of all vertices \(O_{3}^{*}\) are different, \(c\) is a locating-chromatic coloring. Thus, \(\chi_{L}(O_{3}^{*})\leq 4\). **Case B. **\(\chi_{L}(O_{n}^{*})=5\) for \(n\geq 4\)  
Since a subdivision of origami graphs \(O_{n}^{*}\) for \(n\geq 4\) is obtained by origami graph \(O_{n}\) with one added vertex in edge \(v_{i}w_{i}\), we have \(\chi_{L}(O_{n}^{*})\geq 5\) for \(n\geq 4\). The addition of one vertex on the outside does not reduce the number of colors needed because the number of the sets \(B=\{c(u_{i}),c(v_{i}),c(w_{i}),c(u_{i+1})\}\) is the same. To show the upper bound for the locating-chromatic number for a subdivision outer edge of origami graph \(O_{n}^{*}\) for \(n\geq 4\), let us consider different subcases. **Subcase a. ** (odd \(n\)), for \(\left\lceil\frac{n}{2}\right\rceil\) odd, \(n\geq 5\). Let \(c\) be a coloring for a subdivision outer edge of origami graph \(O_{n}^{*}\), for \(\left\lceil\frac{n}{2}\right\rceil\) odd, and \(n\geq 5\); we make the partition \(\Pi\) of \(V(O_{n}^{*})\):  
\(C_{1}=\{w_{i}|1\leq i\leq n\}\);  
\(C_{2}=\{u_{i}|\text{ for odd }i,3\leq i\leq n\}\cup\{v_{i}|\text{ for even }i,2\leq i\leq n-1\}\cup\{x_{i}|\text{ for odd }i,1\leq i\leq n\}\);  
\(C_{3}=\{u_{i}|\text{ for even }i,2\leq i\leq\left\lceil\frac{n}{2}\right\rceil-1\}\cup\{u_{i}| \text{ for even }i,\left\lceil\frac{n}{2}\right\rceil+3\leq i\leq n-1\}\cup\{v_{i}|\text{ for odd }i,1\leq i\leq n\}\cup\{x_{i}|\text{ for even }i,2\leq i\leq n-1\}\);  
\(C_{4}=\{u_{1}\}\);  
\(C_{5}=\{u_{\left\lceil\frac{n}{2}\right\rceil+1}\}\). For for \(\left\lceil\frac{n}{2}\right\rceil\) odd the color codes of all the vertices of \(V(O_{n}^{*})\) are:\[c_{\Pi}(u_{i})=