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A note on the chromatic number of the square of a sparse random graph

Alan Frieze, Aditya Raut

TL;DR

The paper proves that for the square of the sparse random graph $G_{n,p}$ with $p=c/n$, the whp list-chromatic number satisfies $\chi_\ell(G_2)\sim \Delta(G_{n,p})\sim \frac{\log n}{\log\log n}$. It introduces a three-phase greedy list-coloring strategy based on a partition into high-degree vertices $V_\varepsilon$, their neighborhood $W_\varepsilon$, and the remaining vertices, achieving a list size bound $q=(1+3\theta^{1/3})\Delta$ with $\theta=o(1)$. Key auxiliary results (Lemmas 1–5) bound distances, neighbor counts, and edge concentrations in $G_2$, enabling Corollaries that guarantee the three coloring steps succeed, and thereby improve upon previous bounds that $\chi(G_2)\le 6\Delta$. The conclusions discuss potential extensions to larger $p$-ranges, tightness questions, and the behavior of higher powers of $G_{n,p}$ in sparse regimes.

Abstract

We show that w.h.p the list chromatic number $χ_\ell$ of the square of $G_{n,p}$ for $p=c/n$ is asymptotically equal to the maximum degree $Δ(G_{n,p})$. Since $χ(G^2_{n,p})\leq χ_\ell(G^2_{n,p})$, this also improves an earlier result of Garapaty et al \cite{KLMP} who proved that $χ(G^2_{n,p}) \leq 6 \cdot Δ(G_{n,p})$ w.h.p.

A note on the chromatic number of the square of a sparse random graph

TL;DR

The paper proves that for the square of the sparse random graph with , the whp list-chromatic number satisfies . It introduces a three-phase greedy list-coloring strategy based on a partition into high-degree vertices , their neighborhood , and the remaining vertices, achieving a list size bound with . Key auxiliary results (Lemmas 1–5) bound distances, neighbor counts, and edge concentrations in , enabling Corollaries that guarantee the three coloring steps succeed, and thereby improve upon previous bounds that . The conclusions discuss potential extensions to larger -ranges, tightness questions, and the behavior of higher powers of in sparse regimes.

Abstract

We show that w.h.p the list chromatic number of the square of for is asymptotically equal to the maximum degree . Since , this also improves an earlier result of Garapaty et al \cite{KLMP} who proved that w.h.p.
Paper Structure (13 sections, 10 theorems, 23 equations, 3 figures)

This paper contains 13 sections, 10 theorems, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Overview of the proof
  3. Organization of the paper
  4. Proof of Theorem \ref{['th1']}
  5. Bounding the number of -neighbors in
  6. Extra colors required for
  7. List coloring of and proofs of lemmas
  8. Proof of Lemma \ref{['lem1']}
  9. Proof of Lemma \ref{['lem2']}
  10. Proof of Lemma \ref{['lem3']}
  11. Proof of Lemma \ref{['lem4']}
  12. Proof of Lemma \ref{['lem5']}
  13. Conclusions

Key Result

Theorem 1

Let $p=c/n$ where $c>0$ is a constant. Let $G_2$ denote the square of $G_{n,p}$. Then w.h.p., $\chi(G_2)\sim \chi_\ell(G_2)\sim\Delta(G_{n,p})\sim\frac{\log n}{\log\log n}$.

Figures (3)

  • Figure 1: The set of colored vertices in $G_2$ at the end of each step
  • Figure 2: Breadth-First-Search trees after deletions (red nodes are in $V_\varepsilon$).
  • Figure 3: The two types of edges introduced in a set $S$ when squaring graph $G$

Theorems & Definitions (19)

  • Theorem 1
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Corollary 4
  • proof
  • Corollary 5
  • proof
  • Lemma 6
  • Lemma 7
  • ...and 9 more