The enumeration of Hamiltonian paths on random bicubic maps is a very simply stated combinatorial problem that still awaits an exact solution. In this talk, I will present estimates of configuration exponents for the asymptotics of ensembles of such Hamiltonian paths with possible defects, as obtained from extrapolations of exact enumerations for finite sizes. I will then compare these measured exponents with theoretical predictions based on the Knizhnik, Polyakov, Zamolodchikov (KPZ) relations applied to classical dimensions for fully packed loops on the honeycomb lattice. I will show that a naive use of the KPZ relations does not reproduce the measured exponents but that a simple modification of a parameter in their application can eventually correct the observed discrepancy. I will also show that a similar modification is needed to reproduce via the KPZ formulas some exactly known exponents for the closely related problem of fully packed unweighted loops on random planar bicubic maps. This presentation is based on joint work with Philippe Di Francesco, Bertrand Duplantier and Olivier Golinelli.