In this paper we prove the following conjecture by Bollobás and Komlós: For every γ > 0 and integers r ≥ 1 and ∆, there exists β > 0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least ((r − 1)/r + γ )n and H is an r -chromatic graph with n vertices, bandwidth at most βn and maximum degree at most ∆, then G contains a copy of H .

No abstract

One of the first results in graph theory was Dirac's theorem which claims that if the minimum degree in a graph is at least half of the number of vertices, then it contains a Hamiltonian cycle. This result has inspired countless other results all stating that in dense graphs we can find sparse spanning subgraphs. Along these lines, one of the most far-reaching results is the celebrated _Bandwidth Theorem_, proved around 10 years ago by Böttcher, Schacht, and Taraz. It states, rougly speaking, that every $n$-vertex graph with minimum degree at least $\left( \frac{r-1}{r} + o(1)\right) n$ contains a copy of all $n$-vertex graphs $H$ such that $\chi(H) \leq r$, $\Delta (H) = O(1)$, and the bandwidth of $H$ is $o(n)$. This was conjectured earlier by Bollobás and Komlós. The proof is using the Regularity method based on the Regularity Lemma and the Blow-up Lemma. Ever since the Bandwith Theorem came out, it has been open whether one could prove a similar statement for sparse random graphs. In this remarkable, deep paper the authors do just that, they establish sparse random analogues of the Bandwidth Theorem. In particular, the authors show that, for every positive integer $\Delta$, if $p \gg \left(\frac{\log{n}}{n}\right)^{1/\Delta}$, then asymptotically almost surely, every subgraph $G\subseteq G(n, p)$ with $\delta(G) \geq \left( \frac{r-1}{r} + o(1)\right) np$ contains a copy of every $r$-colourable spanning (i.e., $n$-vertex) graph $H$ with maximum degree at most $\Delta$ and bandwidth $o(n)$, provided that $H$ contains at least $C p^{-2}$ vertices that do not lie on a triangle (of $H$). (The requirement about vertices not lying on triangles is necessary, as pointed out by Huang, Lee, and Sudakov.) The main tool used in the proof is the recent monumental sparse Blow-up Lemma due to Allen, Böttcher, Hàn, Kohayakawa, and Person.

The bandwidth theorem [Mathematische Annalen, 343(1):175-205, 2009] states that any n-vertex graph G with minimum degree k−1 k + o(1) n contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs.More precisely, we show that for pwith minimum degree k−1 k +o(1) pn contains all n-vertex k-colourable graphs H with maximum degree ∆, bandwidth o(n), and at least Cp −2 vertices not contained in any triangle. A similar result is shown for sufficiently bijumbled graphs, which, to the best of our knowledge, is the first resilience result in pseudorandom graphs for a rich class of subgraphs. Finally, we provide improved results for H with small degeneracy, which in particular imply a resilience result in G(n, p) with respect to the containment of spanning bounded degree trees for p ≫ log n n 1/3 .

We establish relations between the bandwidth and the treewidth of bounded degree graphs G, and relate these parameters to the size of a separator of G as well as the size of an expanding subgraph of G. Our results imply that if one of these parameters is sublinear in the number of vertices of G then so are all the others. This implies for example that graphs of fixed genus have sublinear bandwidth or, more generally, a corresponding result for graphs with any fixed forbidden minor. As a consequence we establish a simple criterion for universality for such classes of graphs and show for example that for each γ > 0 every n-vertex graph with minimum degree ( 3 4 + γ)n contains a copy of every bounded-degree planar graph on n vertices if n is sufficiently large.