  We study the Bethe approximation for a system of long rigid rods of fixed
length k, with only excluded volume interaction. For large enough k, this
system undergoes an isotropic-nematic phase transition as a function of density
of the rods. The Bethe lattice, which is conventionally used to derive the
self-consistent equations in the Bethe approximation, is not suitable for
studying the hard-rods system, as it does not allow a dense packing of rods. We
define a new lattice, called the random locally tree-like layered lattice,
which allows a dense packing of rods, and for which the approximation is exact.
We find that for a 4-coordinated lattice, k-mers with k>=4 undergo a continuous
phase transition. For even coordination number q>=6, the transition exists only
for k >= k_{min}(q), and is first order.
