  In this short note we study two questions about the existence of subgraphs of
the hypercube $Q_n$ with certain properties. The first question, due to
Erd\H{o}s--Hamburger--Pippert--Weakley, asks whether there exists a bounded
degree subgraph of $Q_n$ which has diameter $n$. We answer this question by
giving an explicit construction of such a subgraph with maximum degree at most
120.
  The second problem concerns properties of $k$-additive spanners of the
hypercube, that is, subgraphs of $Q_n$ in which the distance between any two
vertices is at most $k$ larger than in $Q_n$. Denoting by
$\Delta_{k,\infty}(n)$ the minimum possible maximum degree of a $k$-additive
spanner of $Q_n$, Arizumi--Hamburger--Kostochka showed that
  $$\frac{n}{\ln n}e^{-4k}\leq \Delta_{2k,\infty}(n)\leq 20\frac{n}{\ln n}\ln
\ln n.$$ We improve their upper bound by showing that
  $$\Delta_{2k,\infty}(n)\leq 10^{4k} \frac{n}{\ln n}\ln^{(k+1)}n,$$where the
last term denotes a $k+1$-fold iterated logarithm.
