Extremal diameters of 3-coloring graphs of trees

TL;DR

The paper determines the exact extremal values of the 3-coloring graph diameter for trees on n vertices by introducing balanced labelings and linking diameter to the maximum L1-norm of such labelings. It proves a precise equivalence: the diameter equals the maximum ||h||_1 over balanced labelings, enabling a structural analysis via medians and convexity to identify extremal trees. The path P_n uniquely maximizes the diameter (with an explicit formula), while the star and nearly symmetric double stars minimize it (value floor(3n/2)). This work provides a complete extremal classification for the 3-coloring reconfiguration metric on trees and connects to Hom(T,K3) via the 1-skeleton interpretation.

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.

Figures (1)

• Figure 1: A balanced labeling $h\colon V \to \mathbb{Z}$ with $\|h\|_1 = \lfloor 3n/2 \rfloor + 1$. The numbers shown are the values $h(x)$. This tree on $n=10$ vertices has $|N_2(v)|=3$ and $|N_{\ge 4}(v)|=3$. This satisfies the hypothesis of Lemma \ref{['lem:large-2-neighborhood']} for $k=2$ (since $|N_2(v)| = 3 = \lceil 10/2 \rceil - 4 + 2$ and $|N_{\ge 4}(v)| = 3 \ge 2$). The labeling $h$ assigns $3$ to $k=2$ vertices in $N_{\ge 4}(v)$ and $2$ to all other vertices in $N_{\ge 3}(v)$.

Theorems & Definitions (39)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof