New Bounds and Constructions for Neighbor-Locating Colorings of Graphs

Abstract

AbstractA 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 \(\chi _{NL}(G)\) 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 \(S_i\) such that \(d(x,S_i)\ne d(y,S_i)\). The locating chromatic number \(\chi _{L}(G)\) is the minimum k for which G admits a locating k-coloring. It follows that \(\chi (G)\le \chi _L(G)\le \chi _{NL}(G)\) for any graph G, where \(\chi (G)\) is the usual chromatic number of G.We show that for any three integers p, q, r with \(2\le p\le q\le r\) (except when \(2=p=q<r\)), there exists a connected graph \(G_{p,q,r}\) with \(\chi (G_{p,q,r})=p\), \(\chi _L(G_{p,q,r})=q\) and \(\chi _{NL}(G_{p,q,r})=r\). 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 \(\chi _{NL}(G)=k\) and G is connected (this bound is tight). When G has maximum degree \(\varDelta \), they also showed that a smaller upper-bound on n of order \(k^{\varDelta +1}\) 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.KeywordsColoringNeighbor-locating coloringNeighbor-locating chromatic numberIdentification problemLocation problem