On Hamiltonian Cycles in Claw-Free Cubic Graphs - Mohr Elena; Rautenbach Dieter (Discussiones Mathematicae Graph Theory, 2022)
We show that every claw-free cubic graph of order n at least 8 has at most 2⌊n4⌋{2^{\left\lfloor {{n \over 4}} \right\rfloor }} Hamiltonian cycles, and we also characterize all extremal graphs.