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Sphere packing proper colorings of an expander graph

Honglin Zhu

TL;DR

This work introduces graphical error-correcting codes defined on proper q-colorings of a fixed graph, linking sphere-packing-like constraints to one-sided spectral expansion via the functions $f_{δ,λ,d}(n)$ and $g_{δ,λ,d}(n)$. By constructing explicit expander-based graphs and exploiting probabilistic coloring methods, the authors map out regimes where the maximum code size grows exponentially, remains constant, or falls into a unique regime for the coloring-distance. They present two main exponential-regime constructions—one gadget-based on Ramanujan expanders and a second using random bipartite graphs with recoloring tricks—alongside rigorous constant-regime proofs that cap the growth and a detailed analysis of the unique regime for coloring distance. The results reveal sharp phase transitions in code size as a function of δ and λ, enriching the intersection of coding theory and spectral graph theory and suggesting multiple directions for further study, including rate questions and two-sided expanders. Overall, the paper demonstrates that graphical codes exhibit rich, regime-dependent behavior governed by one-sided expansion and combinatorial coloring structure, with potential algorithmic consequences for colorable expanders.

Abstract

We introduce graphical error-correcting codes, a new notion of error-correcting codes on $[q]^n$, where a code is a set of proper $q$-colorings of some fixed $n$-vertex graph $G$. We then say that a set of $M$ proper $q$-colorings of $G$ form a $(G, M, d)$ code if any pair of colorings in the set have Hamming distance at least $d$. This directly generalizes typical $(n, M, d)$ codes of $q$-ary strings of length $n$ since we can take $G$ as the empty graph on $n$ vertices. We investigate how one-sided spectral expansion relates to the largest possible set of error-correcting colorings on a graph. For fixed $(δ, λ) \in [0, 1] \times [-1, 1]$ and positive integer $d$, let $f_{δ, λ, d}(n)$ denote the maximum $M$ such that there exists some $d$-regular graph $G$ on at most $n$ vertices with normalized second eigenvalue at most $λ$ that has a $(G, M, d)$ code. We study the growth of $f$ as $n$ goes to infinity. We partially characterize the regimes of $(δ, λ)$ where $f$ grows exponentially or is bounded by a constant, respectively. We also prove several sharp phase transitions between these regimes.

Sphere packing proper colorings of an expander graph

TL;DR

This work introduces graphical error-correcting codes defined on proper q-colorings of a fixed graph, linking sphere-packing-like constraints to one-sided spectral expansion via the functions and . By constructing explicit expander-based graphs and exploiting probabilistic coloring methods, the authors map out regimes where the maximum code size grows exponentially, remains constant, or falls into a unique regime for the coloring-distance. They present two main exponential-regime constructions—one gadget-based on Ramanujan expanders and a second using random bipartite graphs with recoloring tricks—alongside rigorous constant-regime proofs that cap the growth and a detailed analysis of the unique regime for coloring distance. The results reveal sharp phase transitions in code size as a function of δ and λ, enriching the intersection of coding theory and spectral graph theory and suggesting multiple directions for further study, including rate questions and two-sided expanders. Overall, the paper demonstrates that graphical codes exhibit rich, regime-dependent behavior governed by one-sided expansion and combinatorial coloring structure, with potential algorithmic consequences for colorable expanders.

Abstract

We introduce graphical error-correcting codes, a new notion of error-correcting codes on , where a code is a set of proper -colorings of some fixed -vertex graph . We then say that a set of proper -colorings of form a code if any pair of colorings in the set have Hamming distance at least . This directly generalizes typical codes of -ary strings of length since we can take as the empty graph on vertices. We investigate how one-sided spectral expansion relates to the largest possible set of error-correcting colorings on a graph. For fixed and positive integer , let denote the maximum such that there exists some -regular graph on at most vertices with normalized second eigenvalue at most that has a code. We study the growth of as goes to infinity. We partially characterize the regimes of where grows exponentially or is bounded by a constant, respectively. We also prove several sharp phase transitions between these regimes.
Paper Structure (13 sections, 25 theorems, 77 equations, 2 figures)

This paper contains 13 sections, 25 theorems, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. Main Results
  4. Outline
  5. Preliminaries
  6. The exponential regime
  7. Construction for \ref{['thm_exp']}
  8. Construction for \ref{['thm_bipartite_exp']}
  9. The constant regime
  10. Proof of \ref{['thm_const']} (1)
  11. Proof of \ref{['thm_const']} (2)
  12. The unique regime
  13. Conclusion and future directions

Key Result

Lemma 1.2

For fixed $n$ and $d$, the functions $f_{\delta, \lambda, d}(n)$ and $g_{\delta, \lambda, d}(n)$ are weakly decreasing in $\delta$ and weakly increasing in $\lambda$.

Figures (2)

  • Figure 1: Schematic diagram for the three regimes when $q = 3$. The curves bounding the regimes may not be precise. Note that the boundary of the exponential regime at $(1/4, 0)$ has a positive slope. The growth of $f$ and $g$ in the central region is unknown. We label the regimes with the function that implies the stronger statement.
  • Figure 2: Illustration of the construction of $G_n$ from $H_n$.

Theorems & Definitions (44)

  • Remark 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • proof : Proof of \ref{['lem_monotone']}
  • Lemma 2.1
  • Lemma 2.2: cheeger1971
  • Lemma 2.3: mcdiarmid1989
  • ...and 34 more