Locally identifying coloring of graphs

TL;DR

This paper introduces the concept of locally identifying coloring in graphs, providing bounds for various graph families and proving NP-completeness for a specific coloring decision problem.

Contribution

It defines the locally identifying coloring, establishes bounds for different graph classes, and proves NP-completeness for a coloring decision problem in bipartite graphs.

Findings

Bounds on $oldsymbol{ ext{chi}}_{lid}$ for planar and perfect graphs

NP-completeness of deciding $oldsymbol{ ext{chi}}_{lid}=3$ in certain bipartite graphs

Introduction of locally identifying coloring concept

Abstract

We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring c of a graph G is said to be locally identifying, if for any adjacent vertices u and v with distinct closed neighborhood, the sets of colors that appear in the closed neighborhood of u and v are distinct. Let $\chi_{lid}(G)$ be the minimum number of colors used in a locally identifying vertex-coloring of G. In this paper, we give several bounds on $\chi_{lid}$ for different families of graphs (planar graphs, some subclasses of perfect graphs, graphs with bounded maximum degree) and prove that deciding whether $\chi_{lid}(G)=3$ for a subcubic bipartite graph $G$ with large girth is an NP-complete problem.