The bandwidth theorem for locally dense graphs

TL;DR

This paper extends the bandwidth theorem to locally dense graphs, showing that such graphs with high minimum degree contain any spanning bounded degree, sublinear bandwidth graph.

Contribution

It proves a version of the bandwidth theorem for locally dense graphs, broadening the class of graphs where the theorem applies.

Findings

Locally dense graphs with minimum degree > (1/2 + o(1))n contain all spanning bounded degree, sublinear bandwidth graphs.

The result generalizes previous theorems from globally dense to locally dense graph settings.

Confirms the conjecture for a wider class of graphs, including locally dense structures.

Abstract

The Bandwidth theorem of B\"ottcher, Schacht and Taraz gives a condition on the minimum degree of an $n$-vertex graph $G$ that ensures $G$ contains every $r$-chromatic graph $H$ on $n$ vertices of bounded degree and of bandwidth $o(n)$, thereby proving a conjecture of Bollob\'as and Koml\'os. In this paper we prove a version of the Bandwidth theorem for locally dense graphs. Indeed, we prove that every locally dense $n$-vertex graph $G$ with $\delta (G) > (1/2+o(1))n$ contains as a subgraph any given (spanning) $H$ with bounded maximum degree and sublinear bandwidth.