This talk concerns numerical semigroups, i.e. cofinite submonoids $S$ of $\mathbb{N}$. Wilf's conjecture (1987) on numerical semigroups $S$ states that $e \cdot \ell \ge c$, where $e$ is the embedding dimension of $S$ and $\ell$ is the number of elements of $S$ which are smaller than its conductor $c$. We shall present the main ideas of a recently published proof of Wilf's conjecture in the particular case $e \ge m/3$, where $m$ is the smallest nonzero element of $S$. Asymptotically, most numerical semigroups seem to satisfy $e \ge m/3$ as the genus goes to infinity. The proof to be presented consists in attaching a certain graph $G$ to $S$ and analysing its properties.