Let $S$ be a closed surface of genus $g \geq 2$. There exist two simple closed curves filling $S$?

Definitions:

Two closed curves $\alpha, \beta$ fill $S$ if they have minimal intersection and $S \setminus (\alpha \cup \beta)$ is a union of topological disks.

Two closed curves $\alpha, \beta$ have minimal intersection if $card(\alpha \cap \beta) \leq card(\alpha^' \cap \beta^')$ for all $\alpha^'$ in the same homotopy class of $\alpha$ and for all $\beta^'$ in the same homotopy class of $\beta$.