  We study the infinite-U Hubbard model on ladders of 2, 4 and 6 legs with
nearest (t) and next-nearest (t') neighbor hoppings by means of the
density-matrix renormalization group algorithm. In particular, we analyze the
stability of the Nagaoka state for several values of t' when we vary the
electron density $(\rho)$ from half-filling to the low-density limit. We build
the two-dimensional phase diagram, where the fully spin-polarized and
paramagnetic states prevail. We find that the inclusion of a non-frustrating
next nearest neighbor hopping stabilizes the fully spin-polarized phase up
until |t'/t|=0.5. Surprisingly, for this value of t', the ground state is fully
spin-polarized for almost any electron density 1 $\gtrsim \rho \gtrsim$ 0,
connecting the Nagaoka state to itinerant ferromagnetism at low density. Also,
we find that the previously found checkerboard insulator phase at t'=0 and
$\rho$=0.75 is unstable against t'.
