2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/28394A Hamiltonian cycle of a graph is a closed path that visits every vertex once and only once. It has been difficult to count the number of Hamiltonian cycles on regular lattices with periodic boundary conditions, e.g. lattices on a torus, due to the presence of winding modes. In this paper, the exact number of Hamiltonian cycles on a random trivalent fat graph drawn faithfully on a torus is obtained. This result is further extended to the case of random graphs drawn on surfaces of an arbitrary genus. The conformational exponent gamma is found to depend on the genus linearly.16 pages, 4 figures, latex2e, uses graphicx and amsfonts. minor correctionsStatistical MechanicsSoft Condensed MatterHigh Energy Physics - LatticeHigh Energy Physics - Theorytext