The Max k-Cut Game: On Stable Optimal Colorings

TL;DR

This paper investigates the stability of optimal solutions in the max k-cut game, demonstrating that they are 7-stable equilibria and analyzing properties of subsets with strong deviations.

Contribution

It introduces an alternative formula for cut value differences and establishes that optimal solutions are 7-stable equilibria in the max k-cut game.

Findings

Optimal solutions are 7-stable equilibria.

Nodes in deviation subsets tend to adopt neighbors' colors.

Deviating subsets induce connected subgraphs.

Abstract

We study the max k-cut game on an undirected and unweighted graph in order to find out whether an optimal solution is also a strong equilibrium. While we do fail to show that, by proving an alternate formula for computing the cut value difference for a strong deviation, we show that optimal solutions are 7-stable equilibria. Furthermore, we prove some properties of minimal subsets with respect to a strong deviation, showing that each of their nodes will deviate towards the color of one of their neighbors and that those subsets induce connected subgraphs.