New bounds and constructions for neighbor-locating colorings of graphs

Abstract

A proper k-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 χ N L (G) is the minimum k for which G admits a neighbor-locating k-coloring. A proper k-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 S i such that d(x, S i ) = d(y, S i ). The locating chromatic number χ L (G) is the minimum k for which G admits a locating k-coloring. It follows that χ(G) ≤ χ L (G) ≤ χ N L (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 G p,q,r with χ(G p,q,r ) = p, χ L (G p,q,r ) = q and χ N L (G p,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 χ N L (G) = k and G is connected (this bound is tight). When G has maximum degree ∆, they also showed that a smaller upper-bound on n of order k ∆+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 k 2a+1 + 2b. Moreover, we describe constructions of such graphs which are close to reaching the bound.