Spanning embeddings of arrangeable graphs with sublinear bandwidth

Julia Böttcher, Anusch Taraz, Andreas Würfl
2015 Random structures & algorithms (Print)  
The Bandwidth Theorem of Böttcher, Schacht, and Taraz [Mathematische Annalen 343 (1), 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 ∆(H) ≤ √ 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 + γ)n contain almost all planar graphs on n vertices as subgraphs. Using
more » ... ues developed by Allen, Brightwell, and Skokan [Combinatorica 33 (2), 125-160] 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.
doi:10.1002/rsa.20593 fatcat:doc423tz7neqfjbiorduj44uey