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

2025-11-21

AI Summary

This paper resolves two classical conjectures—Ajtai–Erdős–Komlós–Szemerédi (1981) and Alon–Krivelevich–Sudakov (1999)—for the class of 3-colorable graphs. Specifically, it establishes that every $K_{t,t,t}$-free graph (i.e., containing no balanced complete 3-partite subgraph) with average degree $d$ admits an independent set of size $(1-o(1))nlog d/d$, and if its maximum degree is $Delta$, then its chromatic number satisfies $chi(G) = O(Delta/logDelta)$. This yields the first nontrivial lower bound on independence number for general 3-colorable graphs, surpassing prior results restricted to nearly bipartite graphs. It also marks the first extension of the AKS chromatic bound to $K_{t,t,t}$-free graphs in 25 years. Methodologically, the proof integrates a novel variant of the Rödl Nibble, Turán-type structural analysis, and randomized greedy constructions—enabling explicit large independent set construction and effective coloring control even in dense, unconstrained graphs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Ajtai, Erdős, Komlós, and Szemerédi conjectured in 1981 that for every graph $F$, every $n$-vertex $F$-free graph of average degree $d$ contains an independent set of size $Ω(n log d / d)$. The largest class of graphs for which this was previously known was established by Alon, Krivelevich, and Sudakov in 1999, who proved it for the so-called almost bipartite graphs, namely subgraphs of $K_{1,t,t}$. We prove the conjecture for all 3-colorable graphs $F$, i.e., subgraphs of $K_{t,t,t}$, representing the first progress on the problem in more than 25 years. 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. A closely related conjecture of Alon, Krivelevich, and Sudakov (1999) asserts that any $F$-free graph of maximum degree at most $Δ$ has chromatic number $O(Δ/ log Δ)$. This was previously known only for almost bipartite graphs (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 also establish this conjecture for all $3$-colorable graphs $F$, representing the first progress toward the conjecture since it was posed.

Problem

• Proving the independent set conjecture for 3-colorable K_{t,t,t}-free graphs
• Establishing chromatic number bounds for 3-colorable F-free graphs
• Extending Shearer's triangle-free graph results to K_{t,t,t}-free graphs

Innovation

• Combining Rödl nibble method with Turán-type results
• Proving independent set bounds for K_{t,t,t}-free graphs
• Establishing chromatic number bounds for 3-colorable graphs