On the Comparison of the Distinguishing Coloring and the Locating Coloring of Graphs

Let G be a simple connected graph. Then, χ L ( G ) and χ D ( G ) will denote the locating chromatic number and the distinguishing chromatic number of G , respectively. In this paper, we investigate a comparison between χ L ( G ) and χ D ( G ) . We prove that χ D ( G ) ≤ χ L ( G ) . Moreover, we determine some types of graphs whose locating and distinguishing chromatic numbers are equal. Specially, we characterize all graphs G of order n with property that χ D ( G ) = χ L ( G ) = k , where k = 3 , n - 2 or n - 1 . In addition, we construct graphs G with χ D ( G ) = n and χ L ( G ) = m for every 4 ≤ n ≤ m .