a Dept. Computa¸c˜ao, Universidade Federal do Cear´a, Fortaleza, Brazil Abstract A lid-coloring (locally identifying coloring) of a graph is a proper coloring such that, for any edge uv where u and v have distinct closed neighborhoods, the set of colors used on vertices of the closed neighborhoods of u and v are also distinct. In this paper we obtain a relation between lid-coloring and a variation, called strong lid- coloring. With this, we obtain linear time algorithms to calculate the lid-chromatic number for some classes of graphs with few P 4 ’s. We also prove that the lid- chromatic number is O(n 1/2-ε )-inapproximable in polinomial time for every ε> 0, unless P=NP. Keywords: Locally identifying coloring, inapproximability, cographs, P 4 -sparse graphs. 1 Introduction Given a vertex v of a graph G, let N (v) be the set of neighbors of v and let N [v]= N (v) ∪{v} be the closed neighborhood of v. A vertex is universal if N [v]= V (G). Given a proper coloring c of V (G), let c(S ) for any S ⊆ V (G) be the set of colors in the vertices of S and let the code of a vertex v be c(N [v]) (the colors in the closed neighborhood of v). A lid-coloring c of G is a proper coloring of V (G) such that any two adjacent vertices with distinct closed neighborhoods have distinct codes. That is, for every edge uv, if N [u] = N [v], then c(N [u]) = c(N [v]). The lid- chromatic number χ lid (G) is the minimum integer k such that there is a lid- coloring of G using k colors. Locally identifying colorings were introduced in 2012 by Esperet et al. [7] and are related to distinguishing colorings [1,2,3] and identifying codes [11]. 1 Email: {nicolasam,rudini}@lia.ufc.br 2 The authors are partially supported by CNPq, CAPES and FUNCAP. Nicolas Martins a,2 Rudini Sampaio a,2 Inapproximability of the lid-chromatic number Available online at www.sciencedirect.com Electronic Notes in Discrete Mathematics 50 (2015) 121–126 1571-0653/© 2015 Elsevier B.V. All rights reserved. www.elsevier.com/locate/endm http://dx.doi.org/10.1016/j.endm.2015.07.021