Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs

Abstract

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.

Spanning embeddings of arrangeable graphs with sublinear bandwidth

“…Hence, as observed in [4], it is a direct consequence of Theorem 1 that large n-vertex graphs G with minimum degree at least ( r(S)−1 r(S) + γ )n contain all graphs from H S (n, ) as subgraphs, which extends results of Kühn, Osthus and Taraz [20] (see also [18]). With the help of Theorem 3 we are now able to say considerably more -namely, that in fact almost all graphs from H S (n) are contained in each such graph G.…”

Local resilience of spanning subgraphs in sparse random graphs

“…Many interesting classes of graphs have sublinear bandwidth, for instance the class of all bounded degree planar graphs (see [5]). Thus, Theorem 1.1 applies to quite a large family of graphs and states that the local resilience of the complete graph with respect to containing all bounded degree, k-colourable spanning subgraphs of sublinear bandwidth is 1/k − o(1).…”

On the Relation of Separability, Bandwidth and Embedding

“…Böttcher [5] and Böttcher et al [6] went further and explored relations of bandwidth with other notions, like separability. We say that an n-vertex graph H is γ-separable if there exists a separator set S ⊂ V (H) with |S| ≤ γn such that every component of H − S has at most γn vertices.…”