The neighbor-locating-chromatic number of pseudotrees

Abstract

Ak-coloringof a graphGis a partition of the vertices ofGintokindependent sets,which are calledcolors. Ak-coloring isneighbor-locatingif any two vertices belongingto the same color can be distinguished from each other by the colors of their respectiveneighbors. Theneighbor-locating chromatic number¿NL(G) is the minimum cardinalityof a neighbor-locating coloring ofG.In this paper, we determine the neighbor-locating chromatic number of paths, cycles,fans and wheels. Moreover, a procedure to construct a neighbor-locating coloring ofminimum cardinality for these families of graphs is given. We also obtain tight upperbounds on the order of trees and unicyclic graphs in terms of the neighbor-locatingchromatic number. Further partial results for trees are also established.