The number of cycles of a given length in dense hamiltonian graphs: proving Hilton's conjecture

A classical theorem of Sheehan in 1977 states that every hamiltonian graph $G$ of order $n$ satisfying $e(G)>\lfloor \frac{n^2}{4}\rfloor +1$ contains at least two cycles of every length $\mathrm{\ell }$, $3\le \mathrm{\ell }\le n$. In the same paper, Sheehan recorded a conjecture of Hilton, which strengthens this conclusion by asserting that such a graph contains at least $n-\mathrm{\ell }+2$ cycles of length $\mathrm{\ell }$ for each $3\le \mathrm{\ell }\le n$. We prove Hilton’s conjecture for all hamiltonian graphs of order at least $440$.