On the extremal number of edges in hamiltonian connected graphs

Assume that n and δ are positive integers with 3 ≤ δ < n . Let h c ( n , δ ) be the minimum number of edges required to guarantee an n -vertex graph G with minimum degree δ ( G ) ≥ δ to be hamiltonian connected. Any n -vertex graph G with δ ( G ) ≥ δ is hamiltonian connected if | E ( G ) | ≥ h c ( n , δ ) . We prove that h c ( n , δ ) = C ( n − δ + 1 , 2 ) + δ 2 − δ + 1 if δ ≤ ⌊ n + 3 × ( n mod 2 ) 6 ⌋ + 1 , h c ( n , δ ) = C ( n − ⌊ n 2 ⌋ + 1 , 2 ) + ⌊ n 2 ⌋ 2 − ⌊ n 2 ⌋ + 1 if ⌊ n + 3 × ( n mod 2 ) 6 ⌋ + 1 < δ ≤ ⌊ n 2 ⌋ , and h c ( n , δ ) = ⌈ n δ 2 ⌉ if δ > ⌊ n 2 ⌋ .