Let c be a proper k-coloring of a connected graph G. Let Π = {S 1 , S 2 , . . . , S k } be the induced partition of V (G) by c, where S i is the partition class having all vertices with color i. The colorIf all vertices of G have distinct color codes, then c is called a locating-coloring of G. The locating-chromatic number of G, denoted by χ L (G), is the smallest k such that G posses a locating k-coloring. Clearly, any graph of order n ≥ 2 has locating-chromatic number k, where 2 ≤ k ≤ n. Characterizing all graphs with a certain locating-chromatic number is a difficult problem. Up to now, all graphs of order n with locating chromatic number 2, n − 1, or n have been characterized. In this paper, we characterize all trees whose locating-chromatic number is 3. We also give a family of trees with locating-chromatic number 4.
The locating chromatic number of a graph is defined as the cardinality of a minimum resolving partition of the vertex set ( ) such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in are not contained in the same partition class. In this case, the coordinate of a vertex V in is expressed in terms of the distances of V to all partition classes. This concept is a special case of the graph partition dimension notion. In this paper we investigate the locating chromatic number for two families of barbell graphs.
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