On cluster-tilting graphs for hereditary categories

Let $\mathcal{H}$ be a connected hereditary abelian category with tilting objects. It is proved that the cluster-tilting graph associated with $\mathcal{H}$ is always connected. As a consequence, we establish the connectedness of the tilting graph for the category $\operatorname{coh}\mathbb{X}$ of coherent sheaves over a weighted projective line $\mathbb{X}$ of wild type. The connectedness of tilting graphs for such categories was conjectured by Happel-Unger, which has immediately applications in cluster algebras. For instance, we deduce that there is a bijection between the set of indecomposable rigid objects of the cluster category $\mathcal{C}_{\mathbb{X}}$ of $\operatorname{coh}\mathbb{X}$ and the set of cluster variables of the cluster algebra $\mathcal{A}_{\mathbb{X}}$ associated with $\operatorname{coh}\mathbb{X}$.