# The Locating-Chromatic Number of Origami Graphs
## 1 Introduction  
The study of the partition dimension of connected graphs was introduced by [PERSON] et al. [1, 2] with the aim of finding a new method for attacking the problem of determining the metric dimension in graphs. The application of these metric dimensions can be seen in the navigation of a robot modeled by a graph [3, 4], solving the problem of chemical data classification, and determining how to represent a set of chemical compounds in such a way that different compounds have different representations [5, 6]. The concept of the locating-chromatic number was first introduced by [PERSON] et al. in 2002, with two obtained graph concepts, namely coloring vertices and partition dimensions of a graph [7]. Finding the locating-chromatic number of a graph is one of the interesting (and un-completely solved) problems of graph theory. Let \(G=(V,E)\) be a connected graph; the distance \(d(x,y)\) between two of its vertices \(x\) and \(y\) is the length of the shortest path between them. Let \(c\) be a proper \(k\)-coloring of \(G\) with color \(\{1,2,...,k\}\), and \(\Pi=\{C_{1},C_{2},...,C_{k}\}\) be a partition of \(V(G)\) that is induced by the coloring \(c\). The color code \(c_{\Pi}(v)\) of \(v\) is the ordered \(k\)-tuple \((d(v,C_{1}),d(v,C_{2}),...,d(v,C_{k})))\), where \(d(v,C_{i})=\min\ \{d(v,x):x\in C_{i}\}\) for any \(i\in\{1,2,3,...,k\}\). If all distinct vertices of \(G\) have distinct color codes, then \(c\) is called a \(k\)-locating coloring of \(G\). The locating-chromatic number denoted by \(\chi_{L}(G)\) is the smallest \(k\) such that \(G\) has a locating \(k\)-coloring. Let \(c\) be a locating \(k\)-coloring on graph \(G(V,E)\). Furthermore, the locating-chromatic number has been determined for a few graph classes; for example, if \(P_{n}\) is a path of order \(n\geq 3\) then the locating-chromatic number is \(3\); for a cycle \(C_{n}\) if \(n\geq 3\) is odd, \(\chi_{L}(C_{n})=3\) was obtained, and if \(n\) is even, \(\chi_{L}(C_{n})=4\) was obtained; for a double star graph \((S_{a,b})\), \(1\leq a\leq b\) and \(b\geq 2\), \(\chi_{L}(S_{ab})=b+1\) was obtained. Let \(\Pi=\{S_{1},S_{2},...,S_{k}\}\) be the partition of \(V(G)\) induced by \(c\). A vertex \(v\in G\) is called a dominant vertex if \(d(v,S_{i})=1\), where \(v\
otin S_{i}\). [PERSON] et al. characterized all graphs of order \(n\) with the locating-chromatic number \(n-1\)[8]. The problem of determining the locating-chromatic number of any general graph is an NP-hard problem [9]. This means that to determine the locating-chromatic number of any given graph, we need a specific algorithm. In 2012, [PERSON] and [PERSON] proposed a procedure to obtain the locating-chromatic number of corona products of two graphs [9]. In2014, [PERSON] obtained the locating-chromatic number of a non-homogeneous amalgamation of stars [10]. Moreover, to determine the locating-chromatic number of disconnected graphs, graphs with dominant vertices and graphs of two components have been discussed in [11; 12; 13]. In 2019, the characterization of the locating chromatic number of powers of paths and a condition (sharp upper and lower bounds) for the locating chromatic number of powers of cycles were discussed [14] (see [15] for a discussion of the necessary and sufficient conditions for a pair of two specific start graphs to be an odd mean graph). [PERSON] et al. determined the locating-chromatic number of some Petersen graphs; \(P(n,1)\) four for odd \(n\geq 3\) or five for even \(n\geq 4\) were obtained [16], and in [17] results were obtained for certain barbell graphs. [PERSON] et al. have succeed in determined the locating-chromatic number of homogeneous lobsters [18]. In [19], [PERSON] obtained the locating-chromatic number for non-homogeneous caterpillar graphs and non-homogeneous firecracker graphs. Next, [PERSON] and [PERSON] in 2018 determined the locating-chromatic number of subdivision firecrackers graphs [20] and in [21] obtained the certain operation of generalized Petersen graphs \(sP(n,1)\).