On embedding well-separable graphs

TL;DR

This paper proves that well-separable graphs with bounded degree can be embedded into larger graphs with high minimum degree, extending to graphs with small band-width, thus advancing understanding of graph embedding conditions.

Contribution

The paper introduces new embedding results for well-separable graphs with bounded degree, including extensions to graphs with small band-width.

Findings

Embedding of well-separable graphs with bounded degree is possible under certain minimum degree conditions.

Extension of embedding results to graphs with small band-width.

Provides explicit bounds and conditions for embedding large graphs.

Abstract

Call a simple graph $H$ of order $n$ well-separable, if by deleting a separator set of size $o(n)$ the leftover will have components of size at most $o(n)$. We prove, that bounded degree well-separable spanning subgraphs are easy to embed: for every $\gamma >0$ and positive integer $\Delta$ there exists an $n_0$ such that if $n>n_0$, $\Delta(H) \le \Delta$ for a well-separable graph $H$ of order $n$ and $\delta(G) \ge (1-{1 \over 2(\chi(H)-1)} + \gamma)n$ for a simple graph $G$ of order $n$, then $H \subset G$. We extend our result to graphs with small band-width, too.