The bandwidth theorem for locally dense graphs - Staden, K; Treglown, A (Cambridge University Press, 2020)
The bandwidth theorem of Böttcher, Schacht, and Taraz [Proof of the bandwidth conjecture of Bollobás and Komlós, Mathematische Annalen, 2009] 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 Bollobás and Komlós [The Blow-up Lemma, Combinatorics, Probability, and Computing, 1999]. 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 δ(G)>(1/2+o(1))n contains as a subgraph any given (spanning) H with bounded maximum degree and sublinear bandwidth.