Solutions to four open problems on quorum colorings of graphs

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.