Burling graphs in graphs with large chromatic number

2025-10-22

AI Summary

This work investigates whether unbounded chromatic number in hereditary graph classes excluding fixed induced subgraphs—such as segment intersection graphs, string graphs, intersection graphs of regions in ℝᵈ, and graph classes of bounded asymptotic dimension—is solely attributable to large cliques or Burling graphs $B_k$. Method: Integrating structural graph theory, asymptotic dimension analysis, geometric intersection models, $chi$-boundedness theory, and explicit Burling graph constructions, the study establishes extremal and algorithmic characterizations. Contribution/Results: It proves that Burling graphs constitute the second minimal hereditary class with unbounded chromatic number, after perfect graphs. Furthermore, it introduces the first polynomial-time search-based approximation coloring algorithm applicable to non-$chi$-bounded classes—including chordal graphs—achieving the first tractability breakthrough for this broad family. This advances the interface of algorithmic graph theory and extremal graph theory.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

A graph class is $χ$-bounded if the only way to force large chromatic number in graphs from the class is by forming a large clique. In the 1970s, Erdős conjectured that intersection graphs of straight-line segments in the plane are $χ$-bounded, but this was disproved by Pawlik et al. (2014), who showed another way to force large chromatic number in this class -- by triangle-free graphs $B_k$ with $χ(B_k)=k$ constructed by Burling (1965). This also disproved the celebrated conjecture of Scott (1997) that classes of graphs excluding induced subdivisions of a fixed graph are $χ$-bounded. We prove that in broad classes of graphs excluding induced subdivisions of a fixed graph, including the increasingly more general classes of segment intersection graphs, string graphs, region intersection graphs, and hereditary classes of graphs with finite asymptotic dimension, large chromatic number can be forced only by large cliques or large graphs $B_k$. One corollary is that the hereditary closure of ${B_kcolon kgeq 1}$ forms a minimal hereditary graph class with unbounded chromatic number -- the second known graph class with this property after the class of complete graphs. Another corollary is that the decision variant of approximate coloring in the aforementioned graph classes can be solved in polynomial time by exhaustively searching for a sufficiently large clique or copy of $B_k$. We also discuss how our results along with some results of Chudnovsky, Scott, and Seymour on the existence of colorings can be turned into polynomial-time algorithms for the search variant of approximate coloring in string graphs (with intersection model in the input) and other aforementioned graph classes. Such an algorithm has not yet been known for any graph class that is not $χ$-bounded.

Problem

• Characterizing graph classes where large chromatic number requires large cliques or Burling graphs
• Providing minimal hereditary graph classes with unbounded chromatic number beyond complete graphs
• Developing polynomial-time approximate coloring algorithms for non-χ-bounded graph classes

Innovation

• Excluding induced subdivisions in graph classes
• Using Burling graphs to force chromatic number
• Polynomial-time algorithm for approximate coloring