A bound for the locating chromatic number of trees

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Let $f$ be a proper $k$-coloring of a connected graph $G$ and$Pi=(V_1,V_2,ldots,V_k)$ be an ordered partition of $V(G)$ intothe resulting color classes. For a vertex $v$ of $G$, the colorcode of $v$ with respect to $Pi$ is defined to be the ordered$k$-tuple $c_{{}_Pi}(v)=(d(v,V_1),d(v,V_2),ldots,d(v,V_k)),$where $d(v,V_i)=min{d(v,x):~xin V_i}, 1leq ileq k$. Ifdistinct vertices have distinct color codes, then $f$ is called alocating coloring. The minimum number of colors needed in alocating coloring of $G$ is the locating chromatic number of $G$,denoted by $Cchi_{{}_L}(G)$. In this paper, we study the locating chromatic numbers of trees.We provide a counter example to a theorem of Gary Chartrand et al.[G. Chartrand, D. Erwin, M.A. Henning, P.J. Slater, P. Zhang, The locating-chromatic number of a graph,Bull. Inst. Combin. Appl. 36 (2002) 89-101]about the locating chromatic numbers of trees. Also, we offer a new bound for the locating chromatic number of trees. Then, by constructing a special family of trees, we show that this bound is best possible.