Bounding the Minimum Size of Vertex Sets Intersecting All Maximum Independent Sets in Graphs Without Large Induced Matchings

Core Concepts

Any graph G without induced matching of size t satisfies hpGq ≤ ωpGq^(3t-3+op(1)), where hpGq is the smallest size of a subset of vertices that intersects every maximum independent set of G.

Abstract

The content discusses the problem of bounding the minimum size of a vertex set that intersects every maximum independent set in a graph, denoted as hpGq. The key highlights are: Bollob´as, Erd˝os and Tuza conjectured that any graph G with linear (in |G|) independence number must have sublinear hpGq, which remains an open problem. Hajebi, Li and Spirkl recently considered bounding hpGq by the clique number ωpGq, and conjectured that for graphs G without induced matching of size t, hpGq ≤ ωpGq^Opt(1). The authors prove this conjecture, showing that hpGq ≤ 10ttωpGq^(3t-3)logωpGq for any graph G without induced matching of size t. The proof uses the Vapnik-Chervonenkis dimension, fractional transversal number, and off-diagonal Ramsey theorem.

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Key Insights Distilled From

Piercing independent sets in graphs without large induced matching

by Jiangdong Ai... at arxiv.org 04-01-2024

https://arxiv.org/pdf/2403.19737.pdf

Deeper Inquiries

How can the techniques used in this paper be extended to other graph classes beyond those without large induced matchings?

The techniques used in the paper, such as analyzing the transversal number of set systems and bounding the VC-dimension of a set system, can be extended to other graph classes by adapting the specific properties of those classes. For example, for graphs with specific forbidden subgraphs or with certain chromatic number constraints, similar methods can be applied to analyze the transversal numbers of corresponding set systems. By identifying key structural properties unique to different graph classes, researchers can tailor the approach to suit the characteristics of those classes and derive similar results.

Can the upper bound on hpGq be further improved, perhaps by using different approaches?

While the upper bound on hpGq obtained in the paper is already significant, further improvements may be possible by exploring alternative approaches. One potential avenue for improvement could involve refining the analysis of the VC-dimension of the set system F or exploring different techniques to bound the transversal number of F. Additionally, considering variations in the definition of hpGq or incorporating insights from related areas of graph theory could lead to tighter upper bounds. Collaborative efforts and interdisciplinary approaches may also offer new perspectives for enhancing the upper bound on hpGq.

What are the implications of bounding hpGq in terms of other important graph theoretic problems, such as the Erdős-Hajnal conjecture?

Bounding hpGq has significant implications for various graph theoretic problems, including the Erdős-Hajnal conjecture. By establishing upper bounds on hpGq for specific graph classes, researchers can gain insights into the structural properties of graphs within those classes. The relationship between hpGq and other graph parameters, such as the clique number or chromatic number, provides valuable information for understanding the complexity and behavior of graphs. In the context of the Erdős-Hajnal conjecture, bounding hpGq contributes to the broader understanding of extremal and structural graph theory, potentially leading to advancements in resolving conjectures and open problems in the field.