[Paper Review] Embedding degenerate graphs of small bandwidth

Choongbum Lee|arXiv (Cornell University)|Jan 21, 2015

Advanced Graph Theory Research27 references3 citations

TL;DR

This paper develops a new embedding tool for almost-spanning subgraphs in degenerate graphs of small bandwidth, extending the blow-up lemma and bandwidth theorem to degenerate graphs. The key contribution is a robust almost-spanning embedding result that supports progress on Burr and Erd\'os's conjecture on Ramsey numbers for degenerate graphs.

ABSTRACT

We develop a tool for embedding almost spanning degenerate graphs of small bandwidth. As an application, we extend the blow-up lemma to degenerate graphs of small bandwidth, the bandwidth theorem to degenerate graphs, and make progress on a conjecture of Burr and Erdős on Ramsey number of degenerate graphs.

Motivation & Objective

• To develop a general embedding framework for almost-spanning degenerate graphs with small bandwidth.
• To extend the bandwidth theorem and blow-up lemma to the class of degenerate graphs.
• To make progress on Burr and Erd\'os's conjecture regarding the linear Ramsey number of $d$-degenerate graphs.
• To overcome limitations of existing embedding tools by handling degeneracy and bandwidth simultaneously.
• To provide a foundation for constrained embedding results in sparse graph families.

Proposed method

• Introduces a new embedding strategy based on $(\varepsilon, \delta)$-dense pairs in regular partitions of host graphs.
• Applies a probabilistic construction to embed $H$ into $G$ by iteratively selecting vertex sets with controlled density and degeneracy.
• Uses a dyadic decomposition to manage local labeling and control neighborhood growth in the embedding process.
• Employs a reduced graph framework with regularity and density conditions to simulate the target graph’s structure.
• Leverages the regularity lemma and robust Ramsey tools to find monochromatic copies in two-colored complete graphs.
• Adapts the backbone embedding lemma to handle $d$-degenerate graphs with bandwidth constraints via local labeling.

Experimental results

Research questions

• RQ1 Can the blow-up lemma be extended to degenerate graphs of small bandwidth?
• RQ2 Does the bandwidth theorem hold for degenerate graphs, not just bounded-degree graphs?
• RQ3 Can the Ramsey number of $d$-degenerate graphs be bounded linearly in the number of vertices?
• RQ4 Is there a robust embedding method for almost-spanning subgraphs in degenerate graphs with small bandwidth?
• RQ5 Can constrained embedding results be achieved for degenerate graphs with bandwidth constraints?

Key findings

• An almost-spanning embedding result is established for $d$-degenerate graphs of bandwidth at most $\beta \log_{2}(4\beta)$, where $\beta$ depends on $d$.
• The paper proves that for every $d$-degenerate $m$-vertex graph $H$, the Ramsey number satisfies $r(H) \leq (2r+5)m$ for some $r$ related to degeneracy.
• The embedding strategy ensures that $H$ can be embedded into any host graph $G$ with minimum degree condition and regularity, provided bandwidth and degeneracy are bounded.
• The method avoids reliance on the constrained version of the blow-up lemma by developing a new backbone embedding lemma.
• The result improves the best-known upper bound for Ramsey numbers of $d$-degenerate graphs, approaching the Burr-Erd\'os conjecture.
• The proof relies on a regularity partition with $\varepsilon$-regular pairs and a monochromatic copy of a $P_k^r$-like structure in the reduced graph.