# Existence of Hamiltonian cycle in a 3-regular $C_n$-free graph

I know that Hamiltonian cycle problem in $$3$$-regular triangle-free graphs is $$NP$$-complete. I would like to know how far we can stretch this result. Observing that a triangle is just $$C_3$$ cycle, what is the longest fixed-length cycle $$C_g$$ such that deciding the existence of Hamiltonian cycle in $$3$$-regular $$C_g$$-free graphs is still $$NP$$-complete?

Have a look at this paper "Not Being (Super)Thin or Solid is Hard: A Study of Grid Hamiltonicity" Theorem 6.1 states that HCP (Hamiltonian Cycle Problem) for planar graph of maximum degree $$3$$ and girth $$g$$ (i.e. $$C_{g-1}$$-free) is $$NP$$-complete, for arbitrary $$g>6$$ (i.e. $$g\geq 6$$ in your notation).
So, in your notation, we are left with the undetermined cases of $$g=4,5$$.
So if you are willing to relax $$3$$-regular to $$\Delta(G)=3$$, then $$g$$ (actually $$g+1$$ for the girth value) can be arbitrarily large.