Locally identifying coloring of graphs

Let G = (V, E) be a graph. Let c : V → N be a vertex-coloring of the vertices of G. For any vertex u, we denote by N[u] its closed neighborhood (u and its adjacent vertices), and for any S ⊆ V , let c(S) be the set of colors that appear on the vertices of S. A proper vertex-coloring c is said to be locally identifying, if for any edge uv, N[u] = N[v] ⇒ c(N[u]) = c(N[v]). Let χlid(G) be the minimum number of colors used by a locally identifying proper 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 with large girth is an NP-complete problem.

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Louis Esperet, Sylvain Gravier, Mickaël Montassier, Pascal Ochem, Aline Parreau
• Keywords: Adjacent Vertices, Closed Neighborhood, CORR 2010, Edge Uv, Education