  We determine information theoretic conditions under which it is possible to
partially recover the alignment used to generate a pair of sparse, correlated
Erd\H{o}s-R\'enyi graphs. To prove our achievability result, we introduce the
$k$-core alignment estimator. This estimator searches for an alignment in which
the intersection of the correlated graphs using this alignment has a minimum
degree of $k$. We prove a matching converse bound. As the number of vertices
grows, recovery of the alignment for a fraction of the vertices tending to one
is possible when the average degree of the intersection of the graph pair tends
to infinity. It was previously known that exact alignment is possible when this
average degree grows faster than the logarithm of the number of vertices.
