  This paper studies the problem of recovering the hidden vertex correspondence
between two edge-correlated random graphs. We focus on the Gaussian model where
the two graphs are complete graphs with correlated Gaussian weights and the
Erd\H{o}s-R\'enyi model where the two graphs are subsampled from a common
parent Erd\H{o}s-R\'enyi graph $\mathcal{G}(n,p)$. For dense graphs with
$p=n^{-o(1)}$, we prove that there exists a sharp threshold, above which one
can correctly match all but a vanishing fraction of vertices and below which
correctly matching any positive fraction is impossible, a phenomenon known as
the "all-or-nothing" phase transition. Even more strikingly, in the Gaussian
setting, above the threshold all vertices can be exactly matched with high
probability. In contrast, for sparse Erd\H{o}s-R\'enyi graphs with
$p=n^{-\Theta(1)}$, we show that the all-or-nothing phenomenon no longer holds
and we determine the thresholds up to a constant factor. Along the way, we also
derive the sharp threshold for exact recovery, sharpening the existing results
in Erd\H{o}s-R\'enyi graphs.
  The proof of the negative results builds upon a tight characterization of the
mutual information based on the truncated second-moment computation and an
"area theorem" that relates the mutual information to the integral of the
reconstruction error. The positive results follows from a tight analysis of the
maximum likelihood estimator that takes into account the cycle structure of the
induced permutation on the edges.
