On the Locating Chromatic Number of Trees

Abstract

Some coloring algorithms gives an upper bound for the locating chromatic number of trees with all the vertices not in an end-path colored by only two colors. That means, a better coloring algorithm could be achieved by optimizing the number of colors used in the end-paths. We provide an estimation of the locating chromatic number of trees using the locating chromatic number of its end-palms. We also study the locating chromatic number of palms, a subdivision of star. We also prove $\chi_L(S_n(k))=\Theta(n^{1/k})$; $\chi_L(S_n(3))=(1+o(1))\sqrt[3]{4n}$; and $\chi_L(O_n)=\left\lceil\log_3\left(\frac{n}{4}\right)\right\rceil+3$.