  For a given cluster-tilted algebra $A$ of tame type, it is proved that
different indecomposable $\tau$-rigid $A$-modules have different dimension
vectors. This is motivated by Fomin-Zelevinsky's denominator conjecture for
cluster algebras. As an application, we establish a weak version of the
denominator conjecture for cluster algebras of tame type. Namely, we show that
different cluster variables have different denominators with respect to a given
cluster for a cluster algebra of tame type. Our approach involves
Iyama-Yoshino's construction of subfactors of triangulated categories. In
particular,we obtain a description of the subfactors of cluster categories of
tame type with respect to an indecomposable rigid object, which is of
independent interest.
