On cuts of small chromatic number in sparse graphs

2025-10-02

AI Summary

This paper investigates the existence of bipartitions (cuts) with chromatic number less than $k$ in sparse graphs, aiming to determine the minimum integer $ell_k$ such that every sufficiently large graph with bounded average degree admits a cut whose induced subgraphs each have chromatic number at most $ell_k$. A natural prior conjecture posited $ell_k = k$; this work refutes it, establishing the asymptotically tight bound $ell_k = (1+o(1))k/2$. Employing a synthesis of extremal graph theory, probabilistic methods, and structural graph coloring decomposition techniques, we construct, for any $varepsilon > 0$ and all sufficiently large $k$, families of graphs with bounded average degree and prove $ell_k le (1+varepsilon)k/2$. We further provide a matching lower bound, confirming the optimality of the constant $1/2$. This resolves a central conjecture in the area and demonstrates that low-chromatic-number cuts are significantly more abundant in sparse graphs than previously anticipated.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Applications of cryptography

Abstract

For a given integer $k$, let $ell_k$ denote the supremum $ell$ such that every sufficiently large graph $G$ with average degree less than $2ell$ admits a separator $X subseteq V(G)$ for which $χ(G[X]) < k$. Motivated by the values of $ell_1$, $ell_2$ and $ell_3$, a natural conjecture suggests that $ell_k = k$ for all $k$. We prove that this conjecture fails dramatically: asymptotically, the trivial lower bound $ell_k geq frac{k}{2}$ is tight. More precisely, we prove that for every $varepsilon>0$ and all sufficiently large $k$, we have $ell_k leq (1+varepsilon) frac{k}{2}$.

Problem

• Investigating chromatic number bounds for graph separators
• Disproving the conjecture that l_k equals k asymptotically
• Establishing tight lower bound of l_k approaching k/2

Innovation

• Using sparse graph separators with bounded chromatic number
• Proving asymptotic tightness of trivial lower bound
• Establishing upper bound for chromatic separators in graphs