Characterizing all trees with locating-chromatic number 3

Abstract

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.

On the Locating Chromatic Number of Certain Barbell Graphs

“…Recently, Asmiati et al [8] obtained the locating chromatic number of the generalized Petersen graph ( , 1) for ≥ 3. Baskoro and Asmiati [9] have characterized all trees with locating chromatic number 3. In [10] all trees of order with locating chromatic number − 1 were characterized, for any integers and , where > + 3 and 2 ≤ < /2.…”

Further Results on Locating Chromatic Number for Amalgamation of Stars Linking by One Path

“…[8] determined the locating chromatic number for corona product of some graphs. Beside that, Asmiati et al [3] characterized graphs containing cycle having locating chromatic number tree and Baskoro et al [7] characterized all trees having locating chromatic number three.…”

Locating chromatic number of banana tree

“…Next, for characterization E.T. Baskoro and Asmiati [5] characterized trees whose locating-chromatic number three. Motivated by this, in this paper we determine the locating-chromatic number of banana tree.…”