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Independent sets and colorings of $K_{t,t,t}$-free graphs

Abhishek Dhawan, Oliver Janzer, Abhishek Methuku

TL;DR

<1-2> The paper addresses coloring and independence in F-free graphs, proving the Alon–Krivelevich–Sudakov/AEKS conjectures for all 3-colorable graphs F (i.e., subgraphs of K_{t,t,t}). It introduces a novel left-local sparsity framework and a two-phase Rödl nibble combined with a Turán-type triangle bound to achieve near-optimal independent-set sizes and near-linear color bounds in terms of Δ/log Δ. The results yield χ(G) = O(t^6 Δ/log Δ) and α(G) ≥ (1−o(1)) n log d / d for K_{t,t,t}-free graphs, with corollaries giving c_1(F) ≥ 1 and c_2(F) = O(|V(F)|^6). These advances substantially strengthen long-standing bounds for triangle-free and related classes and resolve specific cases such as K_{2,2,2} in the 3-colorable family.

Abstract

Alon, Krivelevich, and Sudakov conjectured in 1999 that every $F$-free graph of maximum degree at most $Δ$ has chromatic number $O(Δ/ \log Δ)$. This was previously known only for almost bipartite graphs, that is, for subgraphs of $K_{1,t,t}$ (verified by Alon, Krivelevich, and Sudakov themselves), while most recent results were concerned with improving the leading constant factor in the case where $F$ is almost bipartite. We prove this conjecture for all $3$-colorable graphs $F$, i.e. subgraphs of $K_{t,t,t}$, representing the first progress toward the conjecture since it was posed. A closely related conjecture of Ajtai, Erdős, Komlós, and Szemerédi from 1981 asserts that for every graph $F$, every $n$-vertex $F$-free graph of average degree $d$ contains an independent set of size $Ω(n \log d / d)$. We prove this conjecture in a strong form for all 3-colorable graphs $F$. More precisely, we show that every $n$-vertex $K_{t,t,t}$-free graph of average degree $d$ contains an independent set of size at least $(1 - o(1)) n \log d / d$, matching Shearer's celebrated bound for triangle-free graphs (the case $t = 1$) and thereby yielding a substantial strengthening of it. Our proof combines a new variant of the Rödl nibble method for constructing independent sets with a Turán-type result on $K_{t,t,t}$-free graphs.

Independent sets and colorings of $K_{t,t,t}$-free graphs

TL;DR

<1-2> The paper addresses coloring and independence in F-free graphs, proving the Alon–Krivelevich–Sudakov/AEKS conjectures for all 3-colorable graphs F (i.e., subgraphs of K_{t,t,t}). It introduces a novel left-local sparsity framework and a two-phase Rödl nibble combined with a Turán-type triangle bound to achieve near-optimal independent-set sizes and near-linear color bounds in terms of Δ/log Δ. The results yield χ(G) = O(t^6 Δ/log Δ) and α(G) ≥ (1−o(1)) n log d / d for K_{t,t,t}-free graphs, with corollaries giving c_1(F) ≥ 1 and c_2(F) = O(|V(F)|^6). These advances substantially strengthen long-standing bounds for triangle-free and related classes and resolve specific cases such as K_{2,2,2} in the 3-colorable family.

Abstract

Alon, Krivelevich, and Sudakov conjectured in 1999 that every -free graph of maximum degree at most has chromatic number . This was previously known only for almost bipartite graphs, that is, for subgraphs of (verified by Alon, Krivelevich, and Sudakov themselves), while most recent results were concerned with improving the leading constant factor in the case where is almost bipartite. We prove this conjecture for all -colorable graphs , i.e. subgraphs of , representing the first progress toward the conjecture since it was posed. A closely related conjecture of Ajtai, Erdős, Komlós, and Szemerédi from 1981 asserts that for every graph , every -vertex -free graph of average degree contains an independent set of size . We prove this conjecture in a strong form for all 3-colorable graphs . More precisely, we show that every -vertex -free graph of average degree contains an independent set of size at least , matching Shearer's celebrated bound for triangle-free graphs (the case ) and thereby yielding a substantial strengthening of it. Our proof combines a new variant of the Rödl nibble method for constructing independent sets with a Turán-type result on -free graphs.

Paper Structure

This paper contains 19 sections, 27 theorems, 70 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Notation
  5. Comparison with prior work
  6. Proof overview
  7. Structure of the paper
  8. Preliminaries
  9. Turán-type results for $K_{t,t,t}$-free graphs
  10. Coloring $K_{t, t, t}$-free graphs
  11. Independent sets in $K_{t, t, t}$-free graphs
  12. Proof of Proposition \ref{['prop: two stage nibble']} assuming Lemma \ref{['lemma: lb on n/d and n']}
  13. Tracking $N_i/D_i$ and $N_i$ during the nibble and proving Lemma \ref{['lemma: lb on n/d and n']}
  14. Proof of Proposition \ref{['prop: alg iset']}
  15. Proof of Lemma \ref{['lemma: large I']}: establishing that \ref{['cond: iset lb']} holds with high probability
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $G$ be a $K_{1, t, t}$-free graph of maximum degree at most $\Delta$. Then, $\chi(G) = O\left(\frac{t\Delta}{\log \Delta}\right).$

Figures (2)

  • Figure 1: An edge $e$ contained in $q_e$ copies of $K_3 \cong K_{1, 1, 1}$.
  • Figure 2: A copy $S$ of $K_{1, t}$ contained in $q_S$ copies of $K_{1, 1, t}$.

Theorems & Definitions (50)

  • Conjecture 1.1: Ajtai--Erdős--Komlós--Szemerédi ajtai1981turan
  • Theorem 1.2: Alon--Krivelevich--Sudakov AKSConjecture
  • Conjecture 1.3: Alon--Krivelevich--Sudakov AKSConjecture
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1: Lovász Local Lemma; MolloyReed
  • Theorem 2.2: Chernoff Bound; MolloyReed
  • Theorem 2.3: Talagrand's Inequality; molloy2014colouring
  • Lemma 3.1
  • ...and 40 more