Locating-chromatic number of the edge-amalgamation of trees

The investigation on the locating-chromatic number for graphs was initially studied by Chartrand et al. on 2002. This concept is in fact a special case of the partition dimension for graphs. Even though this topic has received much attention, the current progress is still far from satisfaction. We can define the locating-chromatic number of a graph G as the smallest integer k such that there exists a proper k-coloring on the vertex-set of G such that all vertices have distinct coordinates (color codes) with respect to this coloring. Not like the metric dimension of any tree which is completely solved, the locating-chromatic number for most types of trees are still open. In this paper, we study the locating-chromatic number of trees. In particular, we give lower and upper bounds of the locating-chromatic number of trees formed by an edge-amalgamation of the collection of smaller trees. We also show that the bounds are tight.


Introduction
The topic of locating-chromatic number of graphs was introduced by Chartrand et al. [5] on 2002. They determined the locating-chromatic numbers of some well-known classes of graphs, i.e., paths, cycles, and double stars. They also characterized all graphs of order n with locatingchromatic number n, i.e. multipartite complete graphs. This topic has received much attention. Inspired by Chartrand et al., other authors have determined the locating-chromatic numbers of some well-known classes of graphs. But the results are still limited. In particular for trees, the locating-chromatic number for most types of trees are still open. Some classes of trees with their locating-chromatic numbers known are amalgamations of stars and firecrackers by Asmiati et al. [1,2], homogeneous lobsters and binary trees by Syofyan et al. [6,7], and complete n-arry trees by Welyyanti et al. [9]. Furthermore, all trees on n vertices with locating-chromatic number 3 or n − t where 2 ≤ t < n 2 have been successfully characterized, see [4] and [8], respectively. In this paper, our aim is to determine the locating-chromatic number of the edge-amalgamation of trees. We then estimate the locating-chromatic numbers for some structures of trees obtained by the edge-amalgamation of trees.
Throughout this paper, we only deal with connected graphs. Let G = (V, E) be a connected graph. For u, v ∈ V (G), let d(u, v) denote the distance between u and v. A k-coloring of G is a function c : V (G) → {1, 2, . . . , k} such that c(u) = c(v) for any two adjacent vertices u and v. In other words, c is a partition Π of V (G) into color classes C 1 , C 2 , . . . , C k , where the vertices of C i are colored by i for If any two distinct vertices of G have distinct color codes, then c is called a locating k-coloring of G. Moreover, the least integer k such that there is a locating-coloring in G is called the locating-chromatic number of G, denoted by χ L (G).
The following two results are natural consequences and showed in [5]. Corollary 1.1. If G is a connected graph containing a vertex adjacent to k leaves of G, then χ L (G) ≥ k + 1.

Main Results
For i = 1, 2, . . . , t, let T i be a tree with a fixed edge e o i called the terminal edge. The edgeamalgamation of all these trees T i s, denoted by Edge-Amal{T i ; e o i }, is a tree formed by taking all these trees T i s and identifying their terminal edges. In this section, we will derive the (lower and upper) bounds for the locating-chromatic number of the edge-amalgamation of trees.
Let T be a tree. A stem is a vertex in T that is adjacent to a leaf. A pendant edge is an edge in T incident to a leaf in a tree. For any vertices u and v in T , we denote by u P v the unique path connecting u and v. Let u ∈ V (T ) and define For i = 1, 2, . . . , t, let T i be a tree with a chosen terminal edge e o i = s i l i , where s i is a stem and l i is a leaf. For any stem z of a tree T i we denote N p (z) is the set of pendant vertices adjacent to stem z. Let m i be the number of pendant edges adjacent to stem s i and r i = max{|N p (z)|z is a stem of T i }. Next, in Edge-Amal{T i ; e o i }, we denote s = s i and l = l i .

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Locating-chromatic number of the edge-... | H. Assiyatun, D.K. Syofyan, and E.T. Baskoro Since the coloring c preserves the locating coloring in every tree T 1 , T 2 , . . . , T t , two vertices u and v where c(u) = c(v) and c(N (u)) = c(N (v)) only occur for two cases below.
Then, their color codes are distinguished by the k i -locating coloring c i of T i . Therefore, these vertices are also distinguished by c.
Since c i is a k i -locating coloring and by the definition of the coloring c, there exists integer p = 1, 2 such that c(x) = p for some x ∈ N 2 (s) and x ∈ T i . Thus, we have: and Similarly, consider the subtree T j . Since c j is a k j -locating coloring and by the definition of the coloring c, there exists integer q = 1, 2 and q = p such that c(y) = q for some y ∈ N 2 (s) and y ∈ T j . Thus, we have: and (1), (2), (3) and (4), we have that: Thus, we have that d T (u, C q ) = d T (v, C q ). Therefore, the color codes of u and v are different. A similar argument holds for the case c(u) = c(v) = 2.
Thus, all vertices of the Edge-Amal{T i ; e o i } have distinct color codes. We conclude that

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Locating-chromatic number of the edge-... | H. Assiyatun, D.K. Syofyan, and E.T. Baskoro Next, since there is a stem adjacent to max{r i , 1 + t i=1 (m i − 1)} leaves, by Corollary 1.1 The following two theorems show the existence of trees formed by an edge-amalgamation operation with the locating-chromatic number equals to the lower or upper bounds of Theorem 2.1. Furthermore, in Theorem 2.4, we give the example of trees formed by an edge-amalgation operation with the locating-chromatic number lies in between upper and lower bounds of Theorem 2.1.
Proof. By using the locating-coloring c in proof Theorem 2.1, we have . So, we conclude that Let G w i be a tree having a pendant e o i as depicted in Figure 1, where w i ≥ 2.
We denote x i , y i , z ij the non stem vertex, the stem adjacent to w i leaves, and all leaves adjacent to y i , respectively.
By this coloring, any two vertices u and v satisfying c(u) = c(v) and c(N (u)) = c(N (v)) only occur for the pair of vertices s and y i for w 1 = 2, and the pair of vertices l and x 1 . Their color codes are distinguished by the last ordinate (their distances to a vertex in the color class r + 1). Hence, all vertices have distinct color codes. So, χ L (Edge-Amal(T i ; e o i )) ≤ max{r i + 1}.
Let H m be a tree having a pendant e o i as depicted in Figure 2, where m ≥ 3.
m,j for 1 ≤ k ≤ t. Note that all the colors above in modulo m + 2. We will show that χ L (Edge-Amal(T i ; e o i )) ≤ m + 2. From Theorem 2.3, we shows the exact value of locating-chromatic number for some classes of trees. First, we give definition of some classes of trees and their locating-chromatic number, i.e. double stars, homogeneous caterpillars, and homogeneous lobsters. A double star, denoted by S m,n where n ≥ m ≥ 1, is the graph consisting of two stars K 1,n and K 1,m together with an edge joining their centers. Chartrand et al. [5] have proved χ L (S m,n ) = n + 1. The homogeneous caterpillar C(m, n) is the graph consisting of m stars K 1,n by linking the centers from each stars. Asmiati et al. [3] showed that the locating-chromatic number of homogeneous caterpillar is n + 1 for 1 ≤ m ≤ n + 1, and n + 2 for m > n + 1. The homogeneous lobster Lb(m, n) is the graph obtained by attaching the centers of stars K 1,n to each leaf of C(m, n). Syofyan et al. [6] showed that the locating-chromatic number of the homogeneous lobster is n + 1 if m = 1, n + 2 for 2 ≤ m ≤ 3(n = 2) + 1, or n + 3 for m > 3(n + 2) + 1.
Based on Theorem 2.3 and the locating-chromatic numbers of double stars, homogeneous caterpillars, and homogeneous lobsters, we have the locating-chromatic number of edge-amalgamation of these trees as follows. The terminal edge in each tree is chosen from the edges incident to a stem having maximum leaves.