On minimum degree in Hamiltonian path graphs

For integers p and s satisfying 2 ? s ? p ? 1, let m(p,s) denote the maximum number of edges in a graph G of order p such that the minimum degree in the hamiltonian path graph of G equals s. The values of m(p, s) are determined for 2 ? s ? p/2 and for (2p ? 2)/3 ? s ? p ? 1, and upper and lower bounds on m(p, s) are obtained for p/2 < s < (2p ? 2)/3.