  A critical dilute O($n$) model on the kagome lattice is investigated
analytically and numerically. We employ a number of exact equivalences which,
in a few steps, link the critical O($n$) spin model on the kagome lattice to
the exactly solvable critical $q$-state Potts model on the honeycomb lattice
with $q=(n+1)^2$. The intermediate steps involve the random-cluster model on
the honeycomb lattice, and a fully packed loop model with loop weight
$n'=\sqrt{q}$ and a dilute loop model with loop weight $n$, both on the kagome
lattice. This mapping enables the determination of a branch of critical points
of the dilute O($n$) model, as well as some of its critical properties. For
$n=0$, this model reproduces the known universal properties of the $\theta$
point describing the collapse of a polymer. For $n\neq 0$ it displays a line of
multicritical points, with the same universal properties as a branch of
critical behavior that was found earlier in a dilute O($n$) model on the square
lattice. These findings are supported by a finite-size-scaling analysis in
combination with transfer-matrix calculations.
