Locating Metric Coloring on The Cherry Blossom, Sun Flower and Closed Dutch Windmill Graphs

Abstract

Locating metric coloring is a variation of metric coloring in graphs that integrates vertex coloring with the uniqueness of metric representations. In this coloring, each vertex in a connected graph  is assigned a color such that the distance vectors to each color class are distinct for every pair of different vertices. Let  be a coloring function (not necessarily proper). The coloring ccc is called a locating metric coloring if, for any two distinct vertices , their distance vectors , so it is obtained represents the partition of vertices by color classes. Thus, for every vertex, the distance vector  are different. Vertices may share the same color, whether adjacent or not, as long as their metric representations are unique. The smallest number of colors required for such a coloring is called the locating metric chromatic number, denoted  T This study focuses on analyzing locating metric coloring for three specific graphs: the Cherry Blossom graph , the Sun Flower graph , and the Closed Dutch Windmill graph . These graphs were chosen due to the absence of prior research on their locating metric coloring properties. The research method combines pattern recognition and a deductive-axiomatic approach. The proof process begins by determining lower bounds, followed by the construction of upper bounds through coloring function analysis. The resulting locating metric chromatic numbers for each graph are then established.