  The size of the giant component in the configuration model, measured by the
asymptotic fraction of vertices in the component, is given by a well-known
expression involving the generating function of the degree distribution. In
this note, we argue that the distribution over small degrees is more important
for the size of the giant component than the precise distribution over very
large degrees. In particular, the tail behavior of the degree distribution does
not play the same crucial role for the size of the giant as it does for many
other properties of the graph. Upper and lower bounds for the component size
are derived for an arbitrary given distribution over small degrees $d\leq L$
and given expected degree, and numerical implementations show that these bounds
are close already for small values of $L$. On the other hand, examples
illustrate that, for a fixed degree tail, the component size can vary
substantially depending on the distribution over small degrees.
