Self-intersecting filling curves on surfaces

Let S-g be a closed and oriented surface of genus g >= 2. A closed curve gamma on S-g is said to fill S-g (or simply be filling), if its complement in the surface is a disjoint union of topological discs. It is assumed that the curve gamma is always in minimal position. To a filling curve, we associate a number b, the number of topological discs in its complement. For b = 1, such a filling curve is called minimally intersecting. We prove that for every b >= 1, there exists a filling curve gamma(b) on S-g whose complement is a disjoint union of b many topological discs. Furthermore, we provide an upper bound on the number of mapping class group orbits of closed curves which fills S-g minimally.