On the comparison of the distinguishing coloring and the locating coloring of graphs

TL;DR

This paper compares the locating and distinguishing chromatic numbers of graphs, establishing inequalities, and characterizes graphs where these numbers are equal, especially when both are 3.

Contribution

It proves that the distinguishing chromatic number is always less than or equal to the locating chromatic number and characterizes graphs with equal values.

Findings

Proved that chi D(G) ; chi L(G) for all graphs G.

Identified conditions for chi D(G) = chi L(G).

Characterized graphs with chi D(G) = chi L(G) = 3.

Abstract

Let G be a simple connected graph. Then chi L(G) and chi D(G) will denote the locating chromatic number and the distinguishing chromatic number of G, respectively. In this paper, we investigate a comparison between chi L(G) and chi D(G). In fact, we prove that chi D(G) \leq chi L(G). Moreover, we determine some types of graphs whose locating and distinguishing chromatic numbers are equal. Specially, we characteristic all graph G with the property that chi D(G)= chi L(G) = 3.