[Paper Review] Bandwidth theorem for sparse graphs

Hao Huang, Choongbum Lee+1 more|arXiv (Cornell University)|May 11, 2010

Limits and Structures in Graph Theory9 citations

TL;DR

This paper extends the bandwidth theorem to sparse random graphs, proving that dense random graphs with minimum degree $(1 - 1/r + \gamma)np$ contain copies of any $r$-chromatic graph $H_0$ with bounded bandwidth, provided $H_0$ is bipartite or has vertices with independent neighborhoods in the non-bipartite case. It establishes an asymptotically tight bound on the number of vertex-disjoint copies of $H_0$ in such graphs.

ABSTRACT

A graph $G$ is said to have extit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever $\{i,j\}$ is an edge of $G$. Recently, Bottcher, Schacht, and Taraz verified a conjecture of Bollobas and Komlos which says that for every positive $r,\Delta,\gamma$, there exists $\beta$ such that if $H$ is an $n$-vertex $r$-chromatic graph with maximum degree at most $\Delta$ which has bandwidth at most $\beta n$, then any graph $G$ on $n$ vertices with minimum degree at least $(1 - 1/r + \gamma)n$ contains a copy of $H$ for large enough $n$. In this paper, we extend this theorem to dense random graphs. For bipartite $H$, this answers an open question of Bottcher, Kohayakawa, and Taraz. It appears that for non-bipartite $H$ the direct extension is not possible, and one needs in addition that some vertices of $H$ have independent neighborhoods. We also obtain an asymptotically tight bound for the maximum number of vertex disjoint copies of a fixed $r$-chromatic graph $H_0$ which one can find in a spanning subgraph of $G(n,p)$ with minimum degree $(1-1/r + \gamma)np$.

Motivation & Objective

• To extend the bandwidth theorem from dense graphs to sparse random graphs, specifically $G(n,p)$.
• To resolve an open question on the containment of bipartite graphs with bounded bandwidth in dense random graphs.
• To identify necessary structural conditions—such as independent neighborhoods—for non-bipartite graphs to be embeddable in sparse random graphs.
• To determine the asymptotically tight maximum number of vertex-disjoint copies of a fixed $r$-chromatic graph $H_0$ in a spanning subgraph of $G(n,p)$ with minimum degree $(1 - 1/r + \gamma)np$.

Proposed method

• Adapts the regularity method and blow-up lemma techniques used in the original bandwidth theorem to the setting of random graphs.
• Applies probabilistic methods to analyze the existence of labeled embeddings of $H$ into $G(n,p)$ with bounded bandwidth.
• Uses the concept of bandwidth to control the labeling of vertices in $H$ so that adjacent vertices are close in position, enabling structured embedding.
• Introduces a condition on independent neighborhoods in non-bipartite graphs to compensate for the sparsity of random graphs.
• Employs extremal graph theory and random graph theory to derive tight bounds on the number of vertex-disjoint copies of $H_0$.
• Combines minimum degree conditions with bandwidth constraints to ensure embedding feasibility in sparse random graphs.

Experimental results

Research questions

• RQ1 Can the bandwidth theorem be extended to sparse random graphs $G(n,p)$, particularly for bipartite graphs?
• RQ2 What additional structural conditions are required for non-bipartite graphs with bounded bandwidth to embed into sparse random graphs?
• RQ3 What is the maximum number of vertex-disjoint copies of a fixed $r$-chromatic graph $H_0$ that can be found in a spanning subgraph of $G(n,p)$ with minimum degree $(1 - 1/r + \gamma)np$?
• RQ4 Is the bound on the number of vertex-disjoint copies of $H_0$ asymptotically tight for such graphs?

Key findings

• For bipartite $H$ with bandwidth at most $\beta n$, any dense random graph with minimum degree $(1 - 1/r + \gamma)np$ contains a copy of $H$ for large $n$, confirming an open question.
• For non-bipartite $H$, the embedding is possible only if some vertices of $H$ have independent neighborhoods, indicating a structural necessity beyond bandwidth.
• The maximum number of vertex-disjoint copies of a fixed $r$-chromatic graph $H_0$ in a spanning subgraph of $G(n,p)$ with minimum degree $(1 - 1/r + \gamma)np$ is asymptotically tight.
• The bound on the number of vertex-disjoint copies matches the theoretical maximum, establishing tightness in the random graph setting.
• The results generalize the original bandwidth theorem to sparse random graphs under appropriate structural constraints.