A bound for the locating chromatic number of trees

Let f be a proper k-coloring of a connected graph g and π = (v1, v2, . . . , vk ) be an ordered partition of v (g) into the resulting color classes. for a vertex v of g, the color code of v with respect to π is defined to be the ordered k-tuple cπ (v) = (d(v, v1), d(v, v2), . . . , d(v, vk )), where d(v, vi) = min{d(v, x) : x ∈ vi}, 1 ≤ i ≤ k. if distinct vertices have distinct color codes, then f is called a locating coloring. the minimum number of colors needed in a locating coloring of g is the locating chromatic number of g, denoted by χ (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 number 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.