Embedding into bipartite graphs

TL;DR

This paper improves the minimum degree threshold needed for embedding certain bipartite graphs into large balanced bipartite graphs, leveraging additional structural information of the host graph.

Contribution

It reduces the minimum degree requirement from (1+γ)n to (1/2+γ)n for embedding bounded degree, sublinear bandwidth bipartite graphs, when the host graph is balanced bipartite.

Findings

Threshold halved to (1/2+γ)n for embedding bipartite graphs.

Embedding results complement previous work on K_{r,s}-factors.

Cycle space of the graph is generated by its Hamilton cycles.

Abstract

The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher, 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 $(\frac12+\gamma)n$ 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 Hladk\'y 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.