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 (2 p − 2)/3 ⩽ s ⩽ p − 1, and upper and lower bounds on m ( p, s ) are obtained for p /2 < s < (2 p − 2)/3.