Tight Upper Bounds on Color Reversal by Local Inversions

Abstract

A bicoloration of a graph $G=(V,E)$ is a map $β:V\to\{-1,1\}$. A local inversion at a vertex $v$ complements the subgraph induced by the neighbors of $v$ and simultaneously reverses the colors of all neighbors of $v$. Sabidussi (Discrete Mathematics, 1987) showed that every bicolored graph on $n$ vertices without isolated vertices admits a color reversal using at most $6n+3$ local inversions, and that any two bicolorings of such a graph can be transformed into each other using at most $9n$ local inversions. Recently, Porte, Sandeep, and Santra (CALDAM 2026) improved these bounds to $4n-3$ and $\lfloor(11n-3)/2\rfloor$, respectively. We prove the tight bound $3n$ by showing that, for every graph on $n$ vertices without isolated vertices, any bicoloring can be transformed into any other bicoloring using at most $3n$ local inversions. We also show that this bound is best possible: for complete graphs and stars on $n$ vertices, at least $3n$ local inversions are required to reverse the colors of all vertices. Moreover, the proof of the upper bound is constructive: given two bicolorings, it produces, in polynomial time, a sequence of at most $3n$ local inversions transforming one into the other.