Relaxed Locally Identifying coloring of Graphs

TL;DR

This paper introduces a relaxed version of locally identifying coloring in graphs, explores its computational complexity, provides bounds, and compares it with related graph parameters.

Contribution

It defines the relaxed locally identifying coloring, proves NP-completeness for certain cases, and compares this parameter with existing graph invariants.

Findings

Deciding $ ext{chi}_{rlid}(G)=3$ is NP-complete for 2-degenerate planar graphs.

Provides bounds for $ ext{chi}_{rlid}(G)$ and constructs graphs to tighten these bounds.

Analyzes relationships between $ ext{chi}_{rlid}(G)$, $ ext{chi}_{lid}(G)$, $ ext{gamma}_{id}(G)$, and $ ext{chi}(G)$.

Abstract

A \textit{locally identifying coloring} ($lid$-coloring) of a graph is a proper coloring such that the sets of colors appearing in the closed neighborhoods of any pair of adjacent vertices having distinct neighborhoods are distinct. Our goal is to study a \textit{relaxed locally identifying coloring} ($rlid$-coloring) of a graph that is similar to locally identifying coloring for which the coloring is not necessary proper.We denote by $\chi_{rlid}(G)$ the minimum number of colors used in a relaxed locally identifying coloring of a graph $G$ In this paper, we prove that the problem of deciding that $\chi_{rlid}(G)=3$ for a $2$-degenerate planar graph $G$ is $NP$-complete. We give several bounds of $\chi_{rlid}(G)$ and construct graphs for which some of these bounds are tightened. Studying some families of graphs allows us to compare this parameter with the minimum number of colors used in a locally identifying coloring of a graph $G$ ($\chi_{lid}(G)$), the size of a minimum identifying code of $G$ ($\gamma_{id}(G)$) and the chromatic number of $G$ ($\chi(G)$).