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 \(\chi _{D}(G)\). We prove that \(\chi _{D}(G)\le \chi _L(G)\). 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 \( \chi _{D}(G) = \chi _{L}(G) = k\), where \(k=3,n-2\) or \(n-1\). In addition, we construct graphs G with \(\chi _{D}(G)=n\) and \(\chi _{L}(G)=m\) for every \(4\le n \le m\).
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Acknowledgements
The authors would like to thank to the anonymous referee for the valuable comments and suggestions. The third author has been supported by the program of “Riset Kolaborasi Universitas Top Dunia”, Institut Teknologi Bandung, the Indonesian Ministry of Education, Culture, Research and Technology.
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Korivand, M., Erfanian, A. & Baskoro, E.T. On the Comparison of the Distinguishing Coloring and the Locating Coloring of Graphs. Mediterr. J. Math. 20, 252 (2023). https://doi.org/10.1007/s00009-023-02410-5
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DOI: https://doi.org/10.1007/s00009-023-02410-5