Embedding into bipartite graphs

Abstract

The conjecture of Bollobás and Komlós, recently proved by Böttcher, Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], 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 [to appear in SIAM J. Discrete Math.], as well as Hladký and Schacht [to appear in SIAM J. Discrete Math.], who determined a corresponding minimum degree threshold for $K_{r,s}$-factors, with $r$ and $s$ fixed. Moreover, it implies that the set of Hamilton cycles of $G$ is a generating system for its cycle space.

16 pages, 2 figures

References in corpus (1)

• Tripartite Version of the Corrádi-Hajnal Theorem

Cited by in corpus (2)

• Ramsey numbers for bipartite graphs with small bandwidth
• On prisms, Möbius ladders and the cycle space of dense graphs