On the extremal number of edges in hamiltonian graphs

Abstract

Assume that n and δ are positive integers with 2 ≤ δ < n. Let h(n, δ) be the minimum number of edges required to guarantee an n-vertex graph with minimum degree δ(G) ≥ δ to be hamiltonian, i.e., any n-vertex graph G with δ(G) = δ is hamiltonian if |E(G)| ≥ h(n, δ). We prove that h(n, δ)=C(n - δ, 2)+ δ2 + 1 if δ ≤ (equation).