  We consider the random graph $G_{n, {\bf d}}$ chosen uniformly at random from
the set of all graphs with a given sparse degree sequence ${\bf d}$. We assume
${\bf d}$ has minimum degree at least 4, at most a power law tail, and place
one more condition on its tail. For $k\ge 2$ define $\beta_k(G) = \max e(A, B)
+ k(|A|-|B|) - d(A)$, with the maximum taken over disjoint vertex sets $A, B$.
It is shown that the problem of determining if $G_{n, {\bf d}}$ contains a
Hamilton cycle reduces to calculating $\beta_2(G_{n, {\bf d}})$. If $k\ge 2$
and $\delta\ge k+2$, the problem of determining if $G_{n, {\bf d}}$ contains a
$k$-factor reduces to calculating $\beta_k(G_{n, {\bf d}})$.
