Locally identifying coloring in bounded expansion classes of graphs
Abstract
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 and A. Parreau. Locally identifying coloring of graphs. Electronic Journal of Combinatorics, 19(2), 2012.].
Keywords
Cite
@article{arxiv.1212.5468,
title = {Locally identifying coloring in bounded expansion classes of graphs},
author = {Daniel Gonçalves and Aline Parreau and Alexandre Pinlou},
journal= {arXiv preprint arXiv:1212.5468},
year = {2013}
}