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

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