  Let S $\subseteq$ N be a numerical semigroup with multiplicity m = min(S \
{0}) and conductor c = max(N \ S) + 1. Let P be the set of primitive elements
of S, and let L be the set of elements of S which are smaller than c. A
longstand-ing open question by Wilf in 1978 asks whether the inequality |P||L|
$\ge$ c always holds. Among many partial results, Wilf's conjecture has been
shown to hold in case |P| $\ge$ m/2 by Sammartano in 2012. Using graph theory
in an essential way, we extend the verification of Wilf's conjecture to the
case |P| $\ge$ m/3. This case covers more than 99.999% of numerical semigroups
of genus g $\le$ 45.
