  For a fixed degree sequence $\mathcal{D}=(d_1,...,d_n)$, let $G(\mathcal{D})$
be a uniformly chosen (simple) graph on $\{1,...,n\}$ where the vertex $i$ has
degree $d_i$. In this paper we determine whether $G(\mathcal{D})$ has a giant
component with high probability, essentially imposing no conditions on
$\mathcal{D}$. We simply insist that the sum of the degrees in $\mathcal{D}$
which are not 2 is at least $\lambda(n)$ for some function $\lambda$ going to
infinity with $n$. This is a relatively minor technical condition, and when
$\mathcal{D}$ does not satisfy it, both the probability that $G(\mathcal{D})$
has a giant component and the probability that $G(\mathcal{D})$ has no giant
component are bounded away from $1$.
