  We present an extensive numerical study of the Hubbard model on the doped
AB$_2$ chain, both in the weak coupling and the infinite-U limit. Due to the
special unit cell topology, this system displays a rich variety of phases as
function of hole doping ($\delta$) away from half-filling. Near half-filling,
spiral states develop in the weak coupling regime, while Nagaoka itinerant
ferromagnetism is observed in the infinite-U limit. For higher doping the
system phase-separates before reaching a Mott insulating phase of short-range
RVB states at $\delta=1/3$. Moreover, for $\delta>1/3$ we observe a crossover,
which anticipates the Luttinger liquid behavior for $\delta > 2/3$.
