New Bounds and Constructions for Neighbor-Locating Colorings of Graphs

Abstract

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 ∆, 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.