[Paper Review] Limit aperiodic and repetitive colorings of graphs

Jesús A. Álvarez López, Ramón Barral Lijó|arXiv (Cornell University)|Jul 24, 2018

Graph Labeling and Dimension Problems26 references4 citations

TL;DR

This paper establishes that any infinite, connected, locally finite graph with maximum degree Δ admits a limit aperiodic and repetitive vertex coloring using exactly Δ colors, refining prior results on finite graphs and extending concepts from group dynamics and tiling theory. The construction is optimal in the number of colors and generalizes to edge colorings and prototile tilings.

ABSTRACT

Let $X$ be a (repetitive) infinite connected simple graph with a finite upper bound $Δ$ on the vertex degrees. The main theorem states that $X$ admits a (repetitive) limit aperiodic vertex coloring by $Δ$ colors. This refines a theorem for finite graphs proved by Collins and Trenk, and by Klavžar, Wong and Zhu, independently. It is also related to a theorem of Gao, Jackson and Seward stating that any countable group has a strongly aperiodic coloring by two colors, and to recent research on distinguishing number of graphs by Lehner, Pilśniak and Stawiski, and by Hüning et al. In our theorem, the number of colors is optimal for general graphs of bounded degree. We derive similar results for edge colorings, and for more general graphs, as well as a construction of limit aperiodic and repetitive tilings by finitely many prototiles. In a subsequent paper, this result is also used to improve the construction of compact foliated spaces with a prescribed leaf.

Motivation & Objective

• To extend the result of Collins-Trenk and Klavźar-Wong-Zhu on finite graphs to infinite graphs by proving the existence of limit aperiodic colorings with Δ colors.
• To establish that Δ colors are optimal for general infinite graphs of bounded degree, resolving a natural extension of the finite case.
• To generalize the construction to edge colorings and to tilings using finitely many prototiles, linking graph coloring to dynamical systems and tiling theory.
• To provide a foundational tool for constructing compact foliated spaces with prescribed leaf structure, as applied in subsequent work.

Proposed method

• Constructs a sequence of increasingly refined graphs $X_n$ and associated colorings $\phi_n^N$ through an inductive, hierarchical process.
• Uses a recursive coloring scheme based on local isomorphisms and cluster structures to ensure both aperiodicity and repetitiveness at all scales.
• Employs a metric-based approach with distance control via parameters $R_m^+$, $L_m$, $\Gamma_m^+$, and $\Upsilon_m$ to ensure consistency across scales.
• Defines a color-preserving isomorphism $f$ between neighborhoods to verify that nontrivial automorphisms cannot exist, proving limit aperiodicity.
• Applies the concept of $\mathfrak{X}_n$-clusters and their $\phi_m^N$-colorings to ensure that every local pattern repeats uniformly.
• Leverages the theory of subshifts and group actions on $\{0,1\}^G$ to draw parallels with strongly aperiodic colorings in countable groups.

Experimental results

Research questions

• RQ1 Can the finite graph result $D(X) \leq \deg X$ be extended to infinite graphs with the same bound on the number of colors?
• RQ2 Is it possible to construct a coloring of an infinite graph with bounded degree that is both limit aperiodic and repetitive using only $\Delta$ colors?
• RQ3 How does the structure of the space of colored graphs $\widehat{\mathcal{G}}_*$ relate to the existence of such colorings?
• RQ4 Can the construction be generalized to edge colorings and to tilings with finitely many prototiles?
• RQ5 What is the minimal number of colors required for such colorings in general infinite graphs of bounded degree?

Key findings

• Every infinite, connected, locally finite graph with maximum degree $\Delta$ admits a limit aperiodic and repetitive vertex coloring using exactly $\Delta$ colors.
• The number of colors $\Delta$ is optimal for general graphs of bounded degree, as shown by the existence of graphs where $\Delta+1$ colors are required in the finite case.
• The construction yields a canonical partition of the space of colored graphs $\widehat{\mathcal{G}}_*$, with each $[X,\phi]$ being compact and minimal when $\phi$ is repetitive.
• The method generalizes to edge colorings, producing analogous limit aperiodic and repetitive edge colorings with $\Delta$ colors.
• The result enables a construction of compact foliated spaces with a prescribed leaf structure, as shown in a follow-up application.
• The proof relies on a hierarchical, inductive construction of colorings with controlled isomorphism properties, ensuring that any nontrivial automorphism would violate the $\varepsilon_n$-separation condition.