On locating and neighbor-locating colorings of sparse graphs

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 two sets of colors found in their respective 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)not equal d(y,S-i). The locating chromatic number chi(L)(G) is the minimum k for which G admits a locating k-coloring. Our main results concern the largest possible order of a sparse graph of given neighbor-locating chromatic number. More precisely, we prove that if a connected graph G has order n, neighbor-locating chromatic number k and average degree d, then n is upper-bounded by O(d(2)k((sic)d(sic)+1)). We also design a family of graphs of bounded maximum degree whose order is close to reaching this upper bound. Our upper bound generalizes two previous bounds from the literature, which were obtained for graphs of bounded maximum degree and graphs of bounded cycle rank, respectively. Also, we prove that determining whether chi(L)(G)<= k and chi(NL)(G)<= k are NP-complete for sparse graphs: more precisely, for graphs with average degree at most 7, maximum average degree at most 20 and that are 4-partite. We also study the possible relation between the ordinary chromatic number, the locating chromatic number and the neighbor-locating chromatic number of a graph.