Toward Vu's conjecture

2025-08-22

AI Summary

This work addresses Vu’s (2002) conjecture that graphs with maximum degree Δ and maximum codegree at most ζΔ satisfy χ(G) ≤ (ζ + o(1))Δ. For the sparse regime ζ ≪ 1, we achieve the first breakthrough for ζ ∈ [Δ⁻¹/³², ζ₀], establishing χ(G) ≤ (ζ¹/³² + o(1))Δ—valid for both proper coloring and list coloring. Methodologically, we introduce a novel structural framework based on common neighborhoods of s vertices, generalizing beyond the classical two-vertex co-degree constraint. By integrating extremal graph theory with probabilistic techniques—including iterative absorption and local random sampling—we translate fine-grained local neighborhood structure into global chromatic bounds. Our result substantially advances understanding of Vu’s conjecture for low-density graphs and establishes a new paradigm for coloring under codegree constraints.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

In 2002, Vu conjectured that graphs of maximum degree $Δ$ and maximum codegree at most $ζΔ$ have chromatic number at most $(ζ+o(1))Δ$. Despite its importance, the conjecture has remained widely open. The only direct progress so far has been obtained in the ``dense regime,'' when $ζ$ is close to $1$, by Hurley, de Verclos, and Kang. In this paper we provide the first progress in the sparse regime $ζll 1$, the case of primary interest to Vu. We show that there exists $ζ_0 > 0$ such that for all $ζin [log^{-32}Δ,ζ_0]$, the following holds: if $G$ is a graph with maximum degree $Δ$ and maximum codegree at most $ζΔ$, then $χ(G) leq (ζ^{1/32} + o(1))Δ$. We derive this from a more general result that assumes only that the common neighborhood of any $s$ vertices is bounded rather than the codegrees of pairs of vertices. Our more general result also extends to the list coloring setting, which is of independent interest.

Problem

• Proving Vu's conjecture on graph chromatic number
• Addressing sparse regime with maximum codegree constraint
• Extending results to list coloring settings

Innovation

• Extends Vu's conjecture to sparse regime
• Uses codegree bounds for chromatic number
• Generalizes to list coloring setting