Solutions to four open problems on quorum colorings of graphs

Rafik Sahbi

doi:10.1051/ro/2021116

Abstract

A partition π = {V1, V2,…,Vk} of the vertex set V of a graph G into k color classes Vi, with 1 ≤ i ≤ k is called a quorum coloring if for every vertex v ∈ V, at least half of the vertices in the closed neighborhood N [v] of v have the same color as v. The maximum cardinality of a quorum coloring of G is called the quorum coloring number of G and is denoted ψq (G). In this paper, we give answers to four open problems stated in 2013 by Hedetniemi, Hedetniemi, Laskar and Mulder. In particular, we show that there is no good characterization of the graphs G with ψq (G) nor for those with ψq (G) > 1 unless 𝒫 ≠ 𝒩𝒫 ∩ co – 𝒩𝒫. We also construct several new infinite families of such graphs, one of which the diameter diam (G) of G is not bounded.

Highlights

Let G = (V, E) be a simple graph with order n = |V |
Quorum colorings have several real-world applications, including data clustering, the goal of which is to partition a dataset into homogeneous packets in the sense that the data in the same packet share more characteristics in common between them than with data outside of this packet
This problem can be modeled by a graph G in which each data is represented by a vertex so that two vertices are adjacent if the corresponding data share a fixed minimum number of common characteristics, and the objective is to color the vertex set of the resulting graph such that at least half of the neighbors of each vertex v have the same color as v, where v is counted itself as a neighbor

Summary

Introduction

Quorum colorings have several real-world applications (cf [7, 9] and [12]), including data clustering, the goal of which is to partition a dataset into homogeneous packets in the sense that the data in the same packet share more characteristics in common between them than with data outside of this packet This problem can be modeled by a graph G in which each data is represented by a vertex so that two vertices are adjacent if the corresponding data share a fixed minimum number of common characteristics, and the objective is to color the vertex set of the resulting graph such that at least half of the neighbors of each vertex v have the same color as v, where v is counted itself as a neighbor.

First answer to Question 1

Second answer to Question 1

Answer to Questions 1 and 2

Open problems