Breaking the Bollob\'as-Eldridge-Catlin Barrier for Bipartite Graphs

The celebrated Bollobás-Eldridge-Catlin packing conjecture states that every $n$-vertex graph $G$ with minimum degree at least $\big(1-\frac{1}{\Delta+1}\big) n$ contains every $n$-vertex graph $H$ of maximum degree at most $\Delta$. Despite considerable attention, the conjecture remains widely open.
We show that for bipartite $H$ this threshold can be greatly improved: there is an absolute constant $c>0$ such that every $n$-vertex graph $G$ with minimum degree at least $ \big(1-c\frac{\log\Delta}{\Delta}\big)n $ contains every $n$-vertex bipartite graph $H$ of maximum degree at most $\Delta$, provided $\Delta$ is not too large compared to $n$. Moreover, we prove that this logarithmic improvement is best possible up to the value of the constant.