  Let $\Lambda$ 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 $\Lambda$ form a maximal forward hom-orthogonal sequence of
$\Lambda$-modules whose dimension vectors form the $c$-vectors of a maximal
green sequence for $\Lambda$. Thus we give a proof of Igusa-Todorov's
conjecture.
