MPI-I-2003-1-009

Let $G$ be a simple graph on $n$ vertices. A conjecture of Bollob\'as and Eldridge~\cite{be78} asserts that if $\delta (G)\ge {kn-1 \over k+1}$ then $G$ contains any $n$ vertex graph $H$ with $\Delta(H) = k$. We strengthen this conjecture: we prove that if $H$ is bipartite, $3 \le \Delta(H)$ is bounded and $n$ is sufficiently large , then there exists $\beta >0$ such that if $\delta (G)\ge {\Delta \over {\Delta +1}}(1-\beta)n$, then $H \subset G$.