Locally identifying coloring in bounded expansion classes of graphs

TL;DR

This paper proves that the minimum number of colors needed for a locally identifying coloring is bounded for classes of graphs with bounded expansion, including planar graphs, answering a previously open question.

Contribution

It establishes that the lid-chromatic number is bounded for all classes of bounded expansion, providing explicit bounds for planar graphs and offering alternative proofs for minor closed classes.

Findings

Lid-chromatic number is bounded in bounded expansion classes.

Explicit upper bounds for planar graphs' lid-chromatic number.

Alternative proof for minor closed classes.

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.].