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On the concentration of the chromatic number of random graphs

Erlang Surya, Lutz Warnke

TL;DR

The paper advances the understanding of how the chromatic number of random graphs concentrates across the whole spectrum of edge probabilities. It generalizes Shamir–Spencer and Alon’s results to the sparse regime, proving an interval length of order $\omega\sqrt{n}\,p/\log n$ for a broad range of $p(n)$, using two greedy coloring procedures and Chernoff-type bounds to avoid heavy-deviation techniques. It also uncovers a striking concentration jump in the very dense regime ($p\to1$), proving a concrete polynomial transition near $p=1-n^{-o(1)}$ and, in particular, establishing the $r=1$ case of a conjecture and near-optimal concentration in several ranges. Additionally, the paper identifies a typical value regime where, for certain $p$, $\chi(G_{n,p})=(1+o(1))\,n/r$, complementing the nuanced picture of concentration windows with respect to $p$ and revealing rich, non-monotone behavior as density grows.

Abstract

Shamir and Spencer proved in the 1980s that the chromatic number of the binomial random graph G(n,p) is concentrated in an interval of length at most ω\sqrt{n}, and in the 1990s Alon showed that an interval of length ω\sqrt{n}/\log n suffices for constant edge-probabilities p \in (0,1). We prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case p=p(n) \to 0, and uncover a surprising concentration `jump' of the chromatic number in the very dense case p=p(n) \to 1.

On the concentration of the chromatic number of random graphs

TL;DR

The paper advances the understanding of how the chromatic number of random graphs concentrates across the whole spectrum of edge probabilities. It generalizes Shamir–Spencer and Alon’s results to the sparse regime, proving an interval length of order for a broad range of , using two greedy coloring procedures and Chernoff-type bounds to avoid heavy-deviation techniques. It also uncovers a striking concentration jump in the very dense regime (), proving a concrete polynomial transition near and, in particular, establishing the case of a conjecture and near-optimal concentration in several ranges. Additionally, the paper identifies a typical value regime where, for certain , , complementing the nuanced picture of concentration windows with respect to and revealing rich, non-monotone behavior as density grows.

Abstract

Shamir and Spencer proved in the 1980s that the chromatic number of the binomial random graph G(n,p) is concentrated in an interval of length at most ω\sqrt{n}, and in the 1990s Alon showed that an interval of length ω\sqrt{n}/\log n suffices for constant edge-probabilities p \in (0,1). We prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case p=p(n) \to 0, and uncover a surprising concentration `jump' of the chromatic number in the very dense case p=p(n) \to 1.
Paper Structure (12 sections, 11 theorems, 39 equations, 1 figure)

This paper contains 12 sections, 11 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Concentration bounds for $\chi(G_{n,p})$
  3. Proofs of Theorem and
  4. Proof of Theorem when $zp\gg \log n$
  5. Proof of Theorem when $zp\ll \log n$
  6. Proof of Theorem
  7. Concentration of $\chi(G_{n,p})$: the very dense case
  8. Concentration result: proof of Conjecture for $r=1$
  9. Polynomial concentration jump: proof of Theorem
  10. The typical value
  11. Appendix: proofs of Lemmas --
  12. Appendix: proof of Lemma

Key Result

Theorem 1

Let ${\omega=\omega(n)\to\infty}$ as ${n\to\infty}$ be an arbitrary function, and let ${\delta \in (0,1)}$ be a constant. If the edge-probability ${p=p(n)}$ satisfies ${n^{-1/2+\delta} \ll p\le 1-\delta}$, then there is an interval of length $\lfloor\omega\sqrt{n}p /\log n\rfloor$ that contains the

Figures (1)

  • Figure 1: The exponent of the concentration interval length of very dense random graphs $G_{n,p}$: when ${n^2(1-p)=n^{x+o(1)}}$ with ${x \in (0,2)}$, then Conjecture predicts that ${n^{y+o(1)}}$ with $y=y(x) \in [0,1/2]$ is the length of the shortest interval that contains $\chi(G_{n,p})$ with high probability. Interestingly, this proposes that the concentration interval length of $\chi(G_{n,p})$ has infinitely many polynomial 'jumps' from ${n^{1/2+o(1)}}$ to ${n^{o(1)}}$ as we vary the edge-probability $p=1-n^{-2+x+o(1)}$; see Theorem and Section for more details.

Theorems & Definitions (24)

  • Theorem 1: Improved concentration bound
  • Theorem 2: Concentration 'jump' in the very dense case
  • Theorem 3
  • Theorem 4
  • Lemma 5: Chromatic number: concentration around median
  • Lemma 6: Large independent sets: greedy bound
  • Lemma 7: Chromatic number: greedy bound
  • proof : Proof of Theorem when $zp\gg \log n$
  • proof : Proof of Theorem when $zp\ll \log n$
  • proof : Proof of Theorem
  • ...and 14 more