  We study the statistics of Hamiltonian cycles on various families of
bicolored random planar maps (with the spherical topology). These families fall
into two groups corresponding to two distinct universality classes with
respective central charges $c=-1$ and $c=-2$. The first group includes generic
$p$-regular maps with vertices of fixed valency $p\geq 3$, whereas the second
group comprises maps with vertices of mixed valencies, and the so-called rigid
case of $2q$-regular maps ($q\geq 2$) for which, at each vertex, the unvisited
edges are equally distributed on both sides of the cycle. We predict for each
class its universal configuration exponent $\gamma$, as well as a new universal
critical exponent $\nu$ characterizing the number of long-distance contacts
along the Hamiltonian cycle. These exponents are theoretically obtained by
using the Knizhnik, Polyakov and Zamolodchikov (KPZ) relations, with the
appropriate values of the central charge, applied, in the case of $\nu$, to the
corresponding critical exponent on regular (hexagonal or square) lattices.
These predictions are numerically confirmed by analyzing exact enumeration
results for $p$-regular maps with $p=3,4,\ldots,7$, and for maps with mixed
valencies $(2,3)$, $(2,4)$ and $(3,4)$.
