Maximal forward hom-orthogonal sequences for cluster-tilted algebras of finite type

Let $\mathrm{\Lambda }$ be a cluster-tilted algebra of finite type over an algebraically closed field and $B$ be one of the associated tilted algebras. We show that the $B$-modules, ordered form right to left in the Auslander-Reiten quiver of $\mathrm{\Lambda }$ form a maximal forward hom-orthogonal sequence of $\mathrm{\Lambda }$-modules whose dimension vectors form the $c$-vectors of a maximal green sequence for $\mathrm{\Lambda }$. Thus we give a proof of Igusa-Todorov’s conjecture.