New bounds and constructions for neighbor-locating colorings of graphs
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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_NL(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_NL(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_NL(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_NL(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 k^Δ+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.
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