Quorum colorings of maximum cardinality in linear time for a subclass of perfect trees

¹ Department of the Preparatory Training, National Higher School of Advanced Technologies B.P. 474, Martyrs Square, Algiers 16001, Algeria
² Department of Mathematics, University of Blida 1, Blida, Algeria

^* Corresponding author: r.sahbi@g.essa-alger.edu.dz

Received: 1 August 2025
Accepted: 25 September 2025

Abstract

A partition π = {V₁, V₂, …, V_k} of the vertex set V of a graph G into k color classes V_i, 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 ψ_q(G). A quorum coloring of G of order ψ_q(G) is a ψ_q-coloring of G. In this paper, we design a linear-time algorithm for finding both a ψ_q-coloring and the quorum coloring number for every perfect tree whose the vertices of the same level have the same degree.