The Locating Chromatic Number of Some Graphs

Abstract

A proper coloring of the vertices of a graph G = (V, E) is an assignment of colors to the vertices such that no two adjacent vertices share the same color. The minimum number of colors needed for such a proper coloring is called the chromatic number, denoted by χ(G). In this thesis, we focus on the concept of the locating chromatic number in graph theory, which is a relatively recent refinement of classical vertex coloring. The locating coloring is a proper coloring where any two vertices in the same color class are distinguished by their color codes. The minimum number of colors needed for such a proper coloring is called the locating chromatic number, denoted by χL(G). In this work, we begin by introducing fundamental concepts in graph theory, then study the locating chromatic number by reviewing several existing results related to it. Finally, we present our contributions, providing new results on the locating chromatic number for families of prism graphs namely, the prism itself, the middle graph and the central graph.