Reconfiguration of colorings in triangulations of the sphere

TL;DR

This paper characterizes when 4-colorings of 3-colorable sphere triangulations can be transformed into each other through single-vertex recolorings, extending to higher dimensions and analyzing computational complexity.

Contribution

It provides a characterization and criteria for recoloring sequences in sphere triangulations, generalizes results to higher dimensions, and establishes PSPACE-completeness for related problems.

Findings

Characterization of 4-colorings obtainable from 3-colorings via single-vertex recoloring.

Generalization to k-colorings and (k-2)-spheres for k ≥ 4.

PSPACE-completeness of the recoloring decision problem.

Abstract

In 1973, Fisk proved that any $4$-coloring of a $3$-colorable triangulation of the $2$-sphere can be obtained from any $3$-coloring by a sequence of Kempe-changes. On the other hand, in the case where we are only allowed to recolor a single vertex in each step, which is a special case of a Kempe-change, there exists a $4$-coloring that cannot be obtained from any $3$-coloring. In this paper, we present a characterization of a $4$-coloring of a $3$-colorable triangulation of the $2$-sphere that can be obtained from a $3$-coloring by a sequence of recoloring operations at single vertices, and a criterion for a $3$-colorable triangulation of the $2$-sphere that all $4$-colorings can be obtained from a $3$-coloring by such a sequence. Moreover, our first result can be generalized to a high-dimensional case, in which ``$4$-coloring,'' ``$3$-colorable,'' and ``$2$-sphere'' above are replaced with ``$k$-coloring,'' ``$(k-1)$-colorable,'' and ``$(k-2)$-sphere'' for $k \geq 4$, respectively. In addition, we show that the problem of deciding whether, for given two $(k+1)$-colorings, one can be obtained from the other by such a sequence is PSPACE-complete for any fixed $k \geq 4$. Our results above can be rephrased as new results on the computational problems named {\sc $k$-Recoloring} and {\sc Connectedness of $k$-Coloring Reconfiguration Graph}, which are fundamental problems in the field of combinatorial reconfiguration.