Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sampling Proper Colorings on Line Graphs Using $(1+o(1))Δ$ Colors

Yulin Wang, Chihao Zhang, Zihan Zhang

TL;DR

The paper studies rapid sampling of proper colorings on line graphs via single-site Glauber dynamics. It leverages the matrix trickle-down theorem to translate local spectral information (through links) into global mixing guarantees, exploiting the line-graph structure where each vertex lies in two cliques. By constructing explicit matrix upper bounds and solving a system of inequalities, the authors prove an $O_Δ(n\log n)$ mixing time for line graphs with $q>(1+o(1))Δ$ colors, and obtain refined bounds on spectral gaps and log-Sobolev constants. The work improves upon earlier line-graph results by achieving near-optimal color excess and providing an almost-optimal, explicit construction of the local bounds, enhancing understanding of local-to-global phenomena in high-dimensional expanders. The approach has potential implications for edge-coloring and list-coloring problems in dense local structures beyond line graphs.

Abstract

We prove that the single-site Glauber dynamics for sampling proper $q$-colorings mixes in $O_Δ(n\log n)$ time on line graphs with $n$ vertices and maximum degree $Δ$ when $q>(1+o(1))Δ$. The main tool in our proof is the matrix trickle-down theorem developed by Abdolazimi, Liu and Oveis Gharan (FOCS, 2021).

Sampling Proper Colorings on Line Graphs Using $(1+o(1))Δ$ Colors

TL;DR

The paper studies rapid sampling of proper colorings on line graphs via single-site Glauber dynamics. It leverages the matrix trickle-down theorem to translate local spectral information (through links) into global mixing guarantees, exploiting the line-graph structure where each vertex lies in two cliques. By constructing explicit matrix upper bounds and solving a system of inequalities, the authors prove an mixing time for line graphs with colors, and obtain refined bounds on spectral gaps and log-Sobolev constants. The work improves upon earlier line-graph results by achieving near-optimal color excess and providing an almost-optimal, explicit construction of the local bounds, enhancing understanding of local-to-global phenomena in high-dimensional expanders. The approach has potential implications for edge-coloring and list-coloring problems in dense local structures beyond line graphs.

Abstract

We prove that the single-site Glauber dynamics for sampling proper -colorings mixes in time on line graphs with vertices and maximum degree when . The main tool in our proof is the matrix trickle-down theorem developed by Abdolazimi, Liu and Oveis Gharan (FOCS, 2021).
Paper Structure (24 sections, 28 theorems, 128 equations)

This paper contains 24 sections, 28 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial Complexes
  4. Vertex Coloring
  5. Markov Chains and Mixing Time
  6. Local-to-Global Scheme
  7. Trickle-Down Theorems
  8. Properties of Loewner Order
  9. Vertex Coloring on Line Graphs
  10. Notations
  11. Base Case
  12. Induction Step
  13. Construction of $A_\tau^i$
  14. Spectral analysis of $A_\tau^i$
  15. Construction of $B_\tau$
  16. ...and 9 more sections

Key Result

Theorem 1

Let $G = (V, E)$ be a line graph with $n$ vertices and maximum degree $\Delta$. If $q\ge \Delta + 20028\cdot \Delta / \log\Delta$, then the Glauber dynamics on the $q$-colorings of $G$ has modified log-Sobolev constant $\Omega_\Delta(1/n)$, and thus mixes in time $O_\Delta(n\log n)$.

Theorems & Definitions (49)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Lemma 4: Theorem 12.4 in LPW17
  • Proposition 5: AL20
  • Proposition 6: CLV21
  • Proposition 7: Trickle-Down Theorem in Opp18
  • Proposition 8: Theorem 3.2 in ALOG21
  • Proposition 9: Theorem 1.3 in ALOG21
  • Lemma 10
  • ...and 39 more