A proper k-vertex-coloring of a graph G is a neighbor-locating k-coloring if for each pair of vertices in the same color class, the sets of colors found in their neighborhoods are different. The neighbor-locating chromatic number χNL(G) is the minimum k for which G admits a neighbor-locating k-coloring. A proper k-vertex-coloring of a graph G is a locating k-coloring if for each pair of vertices x and y in the same color-class, there exists a color class Si such that d(x, Si) ≠ d(y, Si). The locating chromatic number χL(G) is the minimum k for which G admits a locating k-coloring. It follows that χ(G) ≤ χL(G) ≤ χNL(G) for any graph G, where χ(G) is the usual chromatic number of G. We show that for any three integers p, q, r with 2 ≤ p≤ q≤ r (except when 2 = p= q< r ), there exists a connected graph Gp,q,r with χ(Gp,q,r) = p, χL(Gp,q,r) = q and χNL(Gp,q,r) = r. We also show that the locating chromatic number (resp., neighbor-locating chromatic number) of an induced subgraph of a graph G can be arbitrarily larger than that of G. Alcon et al. showed that the number n of vertices of G is bounded above by k(2 k-1- 1 ), where χNL(G) = k and G is connected (this bound is tight). When G has maximum degree Δ, they also showed that a smaller upper-bound on n of order kΔ+1 holds. We generalize the latter by proving that if G has order n and at most an+ b edges, then n is upper-bounded by a bound of the order of k2a+1+ 2 b. Moreover, we describe constructions of such graphs which are close to reaching the bound.