Kempe equivalence of $4$-critical planar graphs

TL;DR

This paper proves that all 4-colorings of any 4-critical planar graph are interconnected through Kempe changes, confirming a conjecture related to graph colorings and Kempe equivalence.

Contribution

It establishes that for every 4-critical planar graph, the set of 4-colorings forms a single Kempe class, answering a question posed by Mohar in 2007.

Findings

All 4-colorings of 4-critical planar graphs are Kempe equivalent.

Confirmed a long-standing conjecture in graph coloring theory.

Provides insight into the structure of colorings in planar graphs.

Abstract

Answering a question of Mohar from 2007, we show that for every $4$-critical planar graph, its set of $4$-colorings is a Kempe class.