Extremal diameters of 3-coloring graphs of trees

TL;DR

This paper investigates the maximum and minimum diameters of the 3-coloring graph of trees, introducing balanced labelings to characterize these diameters and identify extremal trees.

Contribution

It introduces the concept of balanced labelings and establishes their connection to the 3-coloring diameter, enabling the determination of extremal values for all trees.

Findings

Maximum 3-coloring diameter for trees on n vertices

Minimum 3-coloring diameter for trees on n vertices

Characterization of extremal trees for diameter values

Abstract

Given a tree $T$, its 3-coloring graph $\mathcal{C}_3(T)$ has as vertices the proper 3-colorings of $T$, with edges joining colorings that differ at exactly one vertex. We call the diameter of $\mathcal{C}_3(T)$ the 3-coloring diameter of $T$. We introduce the notion of balanced labelings of $T$ and show that the 3-coloring diameter equals the maximum $L_1$-norm of a balanced labeling. Using this equivalence, we determine the maximum and minimum values of the 3-coloring diameter over all trees on $n$ vertices and characterize the extremal trees.