Characterization of Graphs with Villainy 2

TL;DR

This paper characterizes graphs for which the maximum villainy, the minimum recoloring needed to fix a permuted optimal coloring without changing color counts, equals two.

Contribution

It provides a complete characterization of graphs with villainy exactly two, expanding understanding of coloring stability under permutations.

Findings

Graphs with villainy 2 are fully characterized.

The characterization helps identify graphs with minimal recoloring complexity.

Results contribute to graph coloring theory and permutation stability.

Abstract

Let $f$ be an optimal proper coloring of a graph $G$ and let $c$ be a coloring of the vertices of $G$ obtained by permuting the colors on vertices in the proper coloring $f$. The villainy of $c$, written $B(c)$, is the minimum number of vertices that must be recolored to obtain a proper coloring of $G$ with the additional condition that the number of times each color is used does not change. The villainy of $G$ is defined as $B(G)=max_{c}B(c)$, over all optimal proper colorings of $G$. In this paper, we characterize graphs $G$ with $B(G)=2$.