Edge bounds in nonhamiltonian kk-connected graphs

Let G be a k-connected graph of order n   with |E(G)|>(n-k2)+k2. Then for (k=1k=1, n⩾3n⩾3), (k=2k=2, n⩾10n⩾10), and (k=3(k=3, n⩾16)n⩾16), GG is hamiltonian. The bounds are tight and for k=1k=1, (k=2k=2, n⩾12n⩾12), and (k=3k=3, n⩾18n⩾18) the extremal graphs are unique. A general bound will also be given for the number of edges in a nonhamiltonian k-connected graph, but the bound is not tight.