Vertex-distinguishing proper edge-colorings
Journal of Graph Theory
Grad and classes with bounded expansion I. Decompositions
European Journal of Combinatorics
Vertex-distinguishing edge colorings of graphs
Journal of Graph Theory
On a new class of codes for identifying vertices in graphs
IEEE Transactions on Information Theory
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A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors used by a locally identifying vertex-coloring. In this paper, we prove that for any graph class of bounded expansion, the lid-chromatic number is bounded. Classes of bounded expansion include minor closed classes of graphs. For these latter classes, we give an alternative proof to show that the lid-chromatic number is bounded. This leads to an explicit upper bound for the lid-chromatic number of planar graphs. This answers in a positive way a question of Esperet et al. [L. Esperet, S. Gravier, M. Montassier, P. Ochem, A. Parreau, Locally identifying coloring of graphs, Electron. J. Combin. 19 (2) (2012)].