# Embedding into bipartite graphs

Böttcher, Julia, Heinig, Peter and Taraz, Anusch (2010) Embedding into bipartite graphs. SIAM Journal on Discrete Mathematics, 24 (4). pp. 1215-1233. ISSN 0895-4801

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## Abstract

The conjecture of Bollobás and Komlós, recently proved by Böttcher, Schacht, and Taraz [Math. Ann., 343 (2009), pp. 175–205], implies that for any $\gamma>0$, every balanced bipartite graph on $2n$ vertices with bounded degree and sublinear bandwidth appears as a subgraph of any $2n$-vertex graph $G$ with minimum degree $(1+\gamma)n$, provided that $n$ is sufficiently large. We show that this threshold can be cut in half to an essentially best-possible minimum degree of $(\frac{1}{2}+\gamma)n$ when we have the additional structural information of the host graph $G$ being balanced bipartite. This complements results of Zhao [SIAM J. Discrete Math., 23 (2009), pp. 888–900], as well as Hladký and Schacht [SIAM J. Discrete Math., 24 (2010), pp. 357–362], who determined a corresponding minimum degree threshold for $K_{r,s}$-factors, with $r$ and $s$ fixed. Moreover, our result can be used to prove that in every balanced bipartite graph $G$ on $2n$ vertices with minimum degree $(\frac{1}{2}+\gamma)n$ and $n$ sufficiently large, the set of Hamilton cycles of $G$ is a generating system for its cycle space.

Item Type: Article http://epubs.siam.org/sidma/ © 2010 Society for Industrial and Applied Mathematics Q Science > QA Mathematics Departments > Mathematics 28 May 2012 15:15 http://eprints.lse.ac.uk/44105/