On the complexity of some quorum colorings problems of graphs

• Main Author: Rafik Sahbi
• Format: Article
• Language: English
• Published: Taylor & Francis Group 2020-09-01
• Series: AKCE International Journal of Graphs and Combinatorics
• Subjects: defensive alliance; quorum colorings; satisfactory partition; cost-effective partition; complexity
• Online Access: http://dx.doi.org/10.1016/j.akcej.2019.12.010

A partition of the vertex set V of a graph G into k color classes with is called a quorum coloring if for every vertex at least half of the vertices in the closed neighborhood of v have the same color as v. The maximum order of a quorum coloring of G is called the quorum coloring number of G and is denoted In this paper, we give answers to two open problems stated in 2013 by Hedetniemi, Hedetniemi, Laskar and Mulder. In fact, we prove that the decision problem associated with is NP-complete when the input graph is a 4-regular graph. We also show that the decision problem asks whether a given graph G has a quorum coloring of order at least 2 is NP-complete too.