Large collections of curves pairwise intersecting exactly once

Abstract

Let $\Omega =(\omega _{j})_{j\in I}$ be a maximum size collection of pairwise non-isotopic simple closed curves on the closed, orientable, genus $g$ surface $S_{g}$, such that $\omega _{i}$ and $\omega _{j}$ intersect exactly once for $i\ne j$. We show that for $g\ge 3$, there exists atleast two such collections up to the action of the mapping class group, answering a question posed by Malestein, Rivin and Theran. As a consequence, we show that the automorphism group of the systole graph for $S_{g}, g\ge 3$ (whose vertices are isotopy classes of simple closed curves, and whose edges correspond to pairs of curve intersecting once) does not act transitively on maximal complete subgraphs.