Quorum colorings of perfect trees

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 of G 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 by ψq(G). A quorum coloring of order ψq(G) is a ψq-coloring. In this paper, we determine the exact value of the quorum coloring number of all perfect trees and show that this value is computable in linear time. Moreover, we design a linear-time algorithm that finds a ψq-coloring for any perfect tree, which partially answers an open question raised by Hedetniemi et al. [AKCE Int. J. Graphs Comb. 10 (2013) 97–109].