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 $\operatorname{ coh}\mathbb{ X}$.
Categories: math.RT math.RA