set of diagonals of ${\mathscr{Z}}$. Then $\add E({\mathscr{X}})$ is a cluster tilting subcategory if and only if ${\mathscr{X}}$ is a maximal set of pairwise non-crossing diagonals, such that for each $a \in L({\mathscr{Z}})$, the set ${\mathscr{X}}$ has a fountain or a leapfrog converging to $a$.

One of the salient features of cluster tilting subcategories are their nice combinatorial properties encoded in the notion of [*cluster structure*]{}. We thank Adam-Christiaan van Roosmalen for pointing out that the following result follows from [@SvR thm. 5.6]. We will give a direct proof.

\[=Theorem \[thm:cluster\_structure\]\] \[thm:C\] The cluster tilting subcategories of $\cC( \cZ )$ form a cluster structure in the sense of [@BIRS sec. II.1].

To get this from [@SvR thm. 5.6] requires the existence of a so-called directed cluster tilting subcategory of $\cC( \cZ )$, which can be obtained from Theorem \[thm:B\] by picking a vertex $z \in \cZ$ and letting $\cX$ be the set of all diagonals from $z$ to non-neighbouring vertices.

Torsion pairs in the cluster categories $\cC( \cZ )$ {#subsec:C}
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Our second main result is a classification of the torsion pairs in $\cC( \cZ )$ (see Definition \[def:torsion\_pairs\]). Recall that torsion pairs in $\cC( A_n )$ correspond to so-called 