Embedding spanning bipartite graphs of small bandwidth

TL;DR

This paper improves conditions under which a large graph contains all bipartite graphs of bounded degree and small bandwidth, using degree sequence and expansion properties.

Contribution

It provides an essentially best-possible degree sequence condition for embedding bipartite graphs of bounded degree and bandwidth o(n) into large graphs.

Findings

Established a degree sequence condition for bipartite graph embedding.

Proved a robust expansion property suffices for embedding.

Confirmed a conjecture on degree sequences and perfect H-packings.

Abstract

Boettcher, Schacht and Taraz gave a condition on the minimum degree of a graph G on n vertices 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 Bollobas and Komlos. We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence of a graph G on n vertices that forces G to contain every bipartite graph H on n vertices of bounded degree and of bandwidth o(n). This also implies an Ore-type result. In fact, we prove a much stronger result where the condition on G is relaxed to a certain robust expansion property. Our result also confirms the bipartite case of a conjecture of Balogh, Kostochka and Treglown concerning the degree sequence of a graph which forces a perfect H-packing.