On the conjectures of neighbor locating coloring of graphs

Let G =(V, E) be a simple graph. Any partition of V(G) to kindependent subsets is called a k-coloring of G. If Pi ={S-1, S-2, . . ., S-k} is a minimum such partition, then k is a chromatic number of G, denoted by chi(G) = k. A k-coloring Pi ={S-1, S-2, . . . , S-k} is said to be a (metric-)locating coloring, (an ML-coloring), if for every pair of distinct vertices u, v, with same color, there exists a color class S-j such that d(u, S-j) not equal d(v, S-j). Minimum kfor ML-coloring of a graph G, is called (metric-)locating chromatic number chi(L)( G) of G. A k-neighbor locating coloring of Gis a partition of V(G) to Pi ={S-1, S-2, . . . , S-k} such that for two vertices u, v is an element of S-i, there is a color class Sjfor which, one of them has a neighbor in Sjand the other not. The minimum kwith this property, is said to be neighbor-locating chromatic number of G, denoted by chi(NL)(G) of G. We initiate to continue the study of neighbor locating coloring of graphs which has been already introduced by other authors. In [1] the authors posed three conjectures and we study these conjectures. We show that, for each pair h,kof integers with 3 <= h <= k, there exists a connected graph Gsuch that chi(L)(G) = hand chi(NL)(G) = k, which proves the first conjecture. If Gand Hare connected graphs, then chi(NL)(G[H]) = chi(NL)(G square H), that disproves the second conjecture. Finally, we investigate for a family of graphs G, chi(NL)(mu(G)) = chi(NL)(G) + 1, where mu(G) is the Mycielski graph of G, that proves the third conjecture for some families of graphs. (C) 2022 Elsevier B.V. All rights reserved.