  We are presented with a graph, $G$, on $n$ vertices with $m$ edges whose edge
set is unknown. Our goal is to learn the edges of $G$ with as few queries to an
oracle as possible. When we submit a set $S$ of vertices to the oracle, it
tells us whether or not $S$ induces at least one edge in $G$. This so-called
OR-query model has been well studied, with Angluin and Chen giving an upper
bound on the number of queries needed of $O(m \log n)$ for a general graph $G$
with $m$ edges.
  When we allow ourselves to make *quantum* queries (we may query subsets in
superposition), then we can achieve speedups over the best possible classical
algorithms. In the case where $G$ has maximum degree $d$ and is
$O(1)$-colorable, Montanaro and Shao presented an algorithm that learns the
edges of $G$ in at most $\tilde{O}(d^2m^{3/4})$ quantum queries. This gives an
upper bound of $\tilde{O}(m^{3/4})$ quantum queries when $G$ is a matching or a
Hamiltonian cycle, which is far away from the lower bound of $\Omega(\sqrt{m})$
queries given by Ambainis and Montanaro.
  We improve on the work of Montanaro and Shao in the case where $G$ has
bounded degree. In particular, we present a randomized algorithm that, with
high probability, learns cycles and matchings in $\tilde{O}(\sqrt{m})$ quantum
queries, matching the theoretical lower bound up to logarithmic factors.
