Relaxed Locally Identifying Coloring of Graphs

Abstract

A locally identifying coloring (lid-coloring) of a graph is a proper vertex-coloring such that the sets of colors appearing in the closed neighborhoods of any pair of adjacent vertices having distinct neighborhoods are distinct. Our goal is to study a relaxed locally identifying coloring (rlid-coloring) of a graph that is similar to locally identifying coloring for which the coloring is not necessarily proper. We denote by $\chi _{rlid}(G)$ the minimum number of colors used in a relaxed locally identifying coloring of a graph G. In this paper, we prove that the problem of deciding that $\chi _{rlid}(G)=3$ for a 2-degenerate planar graph G is NP-complete and for a bipartite graph G is polynomial. We give several bounds of $\chi _{rlid}(G)$ for different families of graphs and construct new graphs for which these bounds are tight. We also compare this parameter with the minimum number of colors used in a locally identifying coloring of a graph G ($\chi _{lid}(G)$), the size of a minimum identifying code of G ($\gamma _{id}(G)$) and the chromatic number of G ($\chi (G)$).