How many simple closed curves can be put on a orientable genus $g$ surface $\Sigma\_g$ such that the following are true:


1. The curves are pairwise non-homotopic
2. The curves are pairwise set-theoretically disjoint.


I am trying to give an upper bound on this number.



Number of non-intersecting non-homotopic simple closed curve
This is Lemma 3.2 of this [paper](http://www.fmf.uni-lj.si/~mohar/Reprints/1996/BM96_JCT68_Juvan_SystemsofCurves.pdf) by Juvan, Malnic and Mohar. 


**Theorem.** Let $F$ be a family of non-null homotopic closed curves on a surface $\Sigma$ with $b \geq 0$ boundary components which are pairwise non-homotopic and pairwise disjoint. Then


$|F| \leq \max (1, 3(g\_{\Sigma}-1)+2b)$,


where $g\_{\Sigma}$ is the genus of $\Sigma$ (orientable or non-orientable).


