On the extremal number of edges in hamiltonian connected graphs

Assume that n and δ are positive integers with 3 ≤ δ < n. Let hc(n, δ) be the minimum number of edges required to guarantee ann-vertex graphGwithminimumdegree δ(G) ≥ δ to be hamiltonian connected. Any n-vertex graphGwith δ(G) ≥ δ is hamiltonian connected if |E(G)| ≥ hc(n, δ). We prove that hc(n, δ) = C(n − δ + 1, 2) + δ2 − δ + 1 if δ ≤ b n+3×(n mod 2) 6 c+1,hc(n, δ) = C(n−b n 2 c+1, 2)+b n 2 c 2 −b n 2 c+1 if b n+3×(n mod 2) 6 c+1 < δ ≤ b n 2 c, and hc(n, δ) = d nδ 2 e if δ > b n 2 c. © 2009 Elsevier Ltd. All rights reserved.