Let ) be a graph with vertex set and edge set For non negative integer a sequence of vertices , such that and for is called a -walk of . If there is no vertex repeated in the sequence, then it is called . For , the length of the shortest -path is called the distance between u and v. The complete graph, denote as , is a graph in which all vertices are adjacent to each other.

Locating-chromatic number was introduced by Chartrand et al. [1], combining the concepts of vertex coloring and partition dimension of a graph. The locating-chromatic number is defined as follows. Let c be a vertex coloring of a connected graph. Define such that for adjacent vertices u and v in . Let be a set of vertices assigned by color i where , defined as color class. Let be an ordered partition of that is induced by coloring c, then the representation of vertex v with respect to is called a color code of denoted as , defined as

where for . Let , vertex v is called a dominant vertex if and for . If all distinct vertices of have distinct color codes, then c is called a k-locating coloring of Then, the locating chromatic number of G is defined as the minimum number of k colors such that has k-locating coloring, denoted by . Chartrand et al.[1] determined the locating-chromatic number of some graph, such as path, cycle, double stars, trees, and multipartite complete graph.

There are some studies about determining locating-coloring number of a graph. In 2013, Purwasih et al.[2] determined the locating-chromatic number for a subdivision of a wheel on one cycle edge. In the same year, Welyyanti, et al.[3] determined the locating chromatic number of homogeneous lobster. Then, in 2014, Behtoei et al.[4] determined the locating chromatic number of the join of graphs, such as the join of path and cycle graph, complete and cycle graph, and two cycles graph. In 2015, Purwasih et al.[5] has determined the bounds on the locating-chromatic number for a subdivision of a graph on one edge. In the same year, Welyyanti et al.[6] determined the locating chromatic number for graphs with dominant vertices. In two years later, Welyyanti et al.[7] also determined the locating-chromatic number for graphs with two homogenous components. In 2021, Irawan et al.[8] has determined the locating-chromatic number of origami graphs. In the same year, Anti et al.[9] determined the locating-chromatic number of the join of path and wheel graph. In the next year, Rahmatalia et al.[10] determined the locating-chromatic number of path split graph, then Sudarsana et al.[11] has also determined the locating chromatic number for m-shadow of a connected graph. Subsequently, in the same year, Fakhri Zikra et al.[12] has determined the locating chromatic number of disjoint union of fan graphs. Then, in 2023, Asmiati et al.[13] determined the locating chromatic number for certain operation of origami graphs. In the same year, Welyyanti et al.[14] determined the locating-chromatic number for certain lobster graph.

In this paper, we study the locating-chromatic number of the sunflower graph for . We achieve this by analyzing the maximum number of color combinations while considering the coloring constraints for each neighbor of a vertex. Using these principles, we establish the exact values for and demonstrate how the graph’s structural properties influence its locating-chromatic number. Furthermore, we outline the methodology used to derive these results and provide insights into the broader implications of our findings. Our work contributes to the understanding of locating-chromatic numbers in structured graph families and ofers directions for further exploration in this area.

Let be a wheel on vertices. Denote the central vertex as o and the vertices on the n-cycle as . The sunflower graph is constructed by adding n vertices , and then adding n edges , edges , and one edge [15]. The sunflower graph has order and size 4n. The sunflower graph will be shown in Figure 1.

The vertex and edge sets of the sunflower graph for are as follows:

(3)

Let c be a locating-coloring in a connected graph . Define such that if u is not adjacent to v. The following theorem gives the locating-chromatic number of the sunflower graph for

Theorem 3.1. Let be a sunflower graph for . Then, the locatingchromatic number of ,

Proof. Consider the following cases:

Case 1.

First, we determine the lower bound of . Since graph is same as graph and graph is a subgraph of , then it is clear that we need at least 4-locating coloring. Consequently, if we use three colors, then three colors will not enough for locating coloring of . As a result, .

Next, we determine the upper bound of . Define , as follows:

The coloring on will be shown in Figure 2.

Then, we have distinct color codes as follows:

Based on the color codes above, all vertices of have distinct color codes.

As a result, . Thus, .

Case 2.

First, we determine the lower bound of for . Assume that there exists 4-locating coloring c on for . Without loss of generality, let . Since o is adjacent to for , then there are at least three colors needed to be assigned to each . The coloring on will be divided into two subcases:

Subcase 2.1. Vertex for will be assigned by one of two colors

Without loss of generality, define the coloring of , as follows:

Vertex for can only be assigned by one of two colors, say color 3 or 4. Let . Since , and , then to avoid two vertices have the same color code, we can assign color 3 for only one vertex, and we assign color 4 to the other vertices. It is clear that and have the same color code because for every . Therefore, we have at least two vertices that have the same color code.

Subcase 2.2. Vertex for will be assigned by one of three colors

By applying the pigeonhole principle, there are at least two vertices that have the same color, say vertices and .

Without loss of generality, let . Now, we can consider three possibilities: either and are both dominant vertices, one of them is dominant vertex, or neither of them are dominant vertices. Then, those will be divided into these subcases, as follows:

Subcase 2.2.1. Either and are both dominant vertices

We can clearly see that both and are dominant vertices. Hence,

Subcase 2.2.2. One of or is dominant vertex

Let be a non-dominant vertex, then we have and . Without loss of generality, let Consequently, we have , , and Since and , then we have some possible colorings for , and If , then or . Alternatively, if , then . Similarly, if , then or . On the other hand, if , then . Now, we can consider three possibilities, as follows:

Subcase 2.2.3. Neither nor are dominant vertices

To demonstrate that this case also contains at least two vertices with the same color code, we follow a similar approach as in Case 2. Considering as a non-dominant vertex, the argument parallels the previous case, wherein we showed that either served as non-dominant vertices. Hence, by adopting the same reasoning, we conclude that there are at least two vertices with identical color codes in this possibility as well.

Based on Subcase 2.1 and Subcase 2.2, we have shown that there are at least two vertices with the same color code. Consequently, this finding contradicts the definition of locating-coloring so four colors are insuficient for locating-coloring on for . Thus, for

Next, we determine the upper bound of for . Assume that there exists 5-locating coloring c on for . Define and be a partition on . Without loss of generality, let , and . Then, we have , and for . Since the color codes of each vertex are distinct, then we can count every possible color code by arranging the coordinates except , as follows:

According to Table 1, consider the first row where , and for . By arranging the coordinate except for , we obtain four possible distinct color codes for , such as , and . It is same for the other rows as well. Then, the number of possible distinct color codes on are 28. Based on the definition of locating-coloring, it is ensured that every vertex has a unique color code. Consequently, since there are 28 possible distinct color codes on from , then there can be a maximum of 28 vertices on and each vertex has a diferent color code. Since the number of is same as the number of in then there can also be a maximum of 28 vertices on . As a result, five colors are still suficient for locating-coloring in when

Based on the upper bound and the lower bound of , we have 5 for .

Consider a 5-locating coloring of . Define , as follows:

The coloring on will be shown in Figure 3

Then, we have distinct color codes, as follows:

Based on the color codes above, all vertices of have diferent color codes. Thus, .

Then, we determine the locating-chromatic number for n \geq 29 by determining the number of possible color codes for q for q that is divided into two cases, as follows.

Case 1. for

Let c be a locating-coloring in . Define and be a partition on . Without loss of generality, let , and . Then, we have , and for . Since the color codes of each vertex are distinct, then we can count every possible color code by arranging the coordinates except , as follows:

According to Table 3, consider the first row where , and for . By arranging the coordinate except for , we obtain five possible distinct color codes for , such as and . It is same for the other rows as well. Then, the number of possible distinct color codes on are 75. Based on the definition of locating-coloring, it is ensured that every vertex has a unique color code. Consequently, since there are possible distinct color codes on from , then there can be a maximum of vertices on and each vertex has a diferent color code. Since the number of is same as the number of in , then there can also be a maximum of 75 vertices on . As a result, six colors are still suficient for locating-coloring in when Based on previous case, since for , then for . Hence, for

Case 2. for

Let c be a locating-coloring in . Define and be a partition on . Without loss of generality, let , and . Then, we have , and for . Since the color codes of each vertex are distinct, then we can count every possible color code by arranging the coordinates except , as follows:

Based on Table 4, consider the first row where , and for . By arranging the coordinate except for , we obtain 6! or 30 possible distinct color codes for . It is same as the other rows as well. The number of possible distinct color codes on is obtained by summing . Then, we have 180 distinct color codes on if Similarly, this holds true for where

Then, consider the number of color codes when , for . By induction, the number of possible distinct color codes on when , are

Based on the definition of locating-coloring, it is ensured that every vertex has a unique color code. Consequently, since there are possible distinct color codes on from , then there can be a maximum of vertices on , and each vertex has a diferent color code. Since the number of is same as the number of in , then there can also be a maximum of vertices on . As a result, q colors for are still suficient for q-locating-coloring in when .

Then, let , for . Based on Equation (4), the maximum number of possible distinct color codes on and are

Therefore, based on Equation (4) and Equation (5), we can conclude that q colors are still suficient for q-locating coloring in if

Thus, we have for

In this paper, we have determined that the locating-chromatic number of the sunflower graph, as follows: