Matroid basis graph: Counting Hamiltonian cycles
  We present exponential and super factorial lower bounds on the number of Hamiltonian cycles passing through any edge of the basis graphs of a graphic, generalized Catalan and uniform matroids. All lower bounds were obtained by a common general strategy based on counting appropriated cycles of length four in the corresponding matroid basis graph. 
Cristina G. Fernandes, C\'esar Hern\'andez-V\'elez, Jos\'e C. de Pina,   and Jorge Luis Ram\'irez Alfons\'in
math.CO