Locally Identifying Coloring of Graphs

We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring c of a graph G is said to be locally identifying, if for any adjacent vertices u and v with distinct closed neighborhood, the sets of colors that appear in the closed neighborhood of u and v are distinct. Let χlid(G) be the minimum number of colors used in a locally identifying vertex-coloring of G. In this paper, we give several bounds on χlid for different families of graphs (planar graphs, some subclasses of perfect graphs, graphs with bounded maximum degree) and prove that deciding whether χlid(G) = 3 for a subcubic bipartite graph G with large girth is an NP-complete problem.