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

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 gamma>0 every n-vertex graph with minimum degree ((3/4)+gamma)n contains a copy of every bounded-degree planar graph on n vertices if n is sufficiently large

Forcing spanning subgraphs via Ore type conditions

Abstract We determine an Ore type condition that allows the embedding of 3-colourable bounded degree graphs of sublinear bandwidth: For all ∆, γ > 0 there are β, n 0 > 0 such that for all n ≥ n 0 the following holds. Let G = (V, E) and H be n-vertex graphs such that H is 3-colourable, has maximum degree ∆(H) ≤ ∆ and bandwidth bw(H) ≤ βn, and G satisfies deg(u) + deg(v) ≥

Local resilience of spanning subgraphs in sparse random graphs

For each real γ>0γ>0 and integers Δ≥2Δ≥2 and k≥1k≥1, we prove that there exist constants β>0β>0 and C>0C>0 such that for all p≥C(log⁡n/n)1/Δp≥C(log⁡n/n)1/Δ the random graph G(n,p)G(n,p) asymptotically almost surely contains – even after an adversary deletes an arbitrary (1/k−γ1/k−γ)-fraction of the edges at every vertex – a copy of every n-vertex graph with maximum degree at most Δ, bandwidth at most βn and at least Cmax⁡{p−2,p−1log⁡n}Cmax⁡{p−2,p−1log⁡n} vertices not in triangles

Spanning embeddings of arrangeable graphs with sublinear bandwidth

The Bandwidth Theorem of B\"ottcher, Schacht and Taraz [Mathematische Annalen 343 (1), 175-205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree. We generalise this result to a-arrangeable graphs H with \Delta(H)<sqrt(n)/log(n), where n is the number of vertices of H. Our result implies that sufficiently large n-vertex graphs G with minimum degree at least (3/4+\gamma)n contain almost all planar graphs on n vertices as subgraphs. Using techniques developed by Allen, Brightwell and Skokan [Combinatorica, to appear] we can also apply our methods to show that almost all planar graphs H have Ramsey number at most 12|H|. We obtain corresponding results for graphs embeddable on different orientable surfaces.Comment: 20 page

Separating a Voronoi Diagram via Local Search

Given a set P of n points in R^dwe show how to insert a set Z of O(n^(1-1/d)) additional points, such that P can be broken into two sets P1 and P2of roughly equal size, such that in the Voronoi diagram V(P u Z), the cells of P1 do not touch the cells of P2; that is, Z separates P1 from P2 in the Voronoi diagram (and also in the dual Delaunay triangulation). In addition, given such a partition (P1,P2) of Pwe present an approximation algorithm to compute a minimum size separator realizing this partition. We also present a simple local search algorithm that is a PTAS for approximating the optimal Voronoi partition

Treewidth of Erdős–Rényi random graphs, random intersection graphs, and scale-free random graphs

AbstractWe study conditions under which the treewidth of three different classes of random graphs is linear in the number of vertices. For the Erdős–Rényi random graph G(n,m), our result improves a previous lower bound obtained by Kloks (1994) [22]. For random intersection graphs, our result strengthens a previous observation on the treewidth by Karoński et al. (1999) [19]. For scale-free random graphs based on the Barabási-Albert preferential-attachment model, it is shown that if more than 11 vertices are attached to a new vertex, then the treewidth of the obtained network is linear in the size of the network with high probability