Let G be a graph, \(\nu \) the order of G and k a positive integer such that \(k\le (\nu -2)/2\). Then G is said to be k-extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G. A graph G is Hamiltonian if it contains a Hamiltonian cycle. A graph G is Hamiltonian-connected if, for any two of its vertices, it contains a spanning path joining the two vertices. In this paper, we discuss k-extendable nonbipartite graphs with \(\kappa (G)\ge 2k+r\) where \(k\ge 1\) and \(r\ge 0\). It is shown that if \(\nu \le 6k+2r\), then G is Hamiltonian; and if \(\nu > 6k+2r\), then G has a longest cycle C such that \(|V(C)|\ge 6k+2r\); and if \(\nu <6k+2r\), then G is Hamiltonian-connected; and if \(\nu \ge 6k+2r\), then for each pair of vertices \(z_1\) and \(z_2\) of G, there is a path P of G joining \(z_1\) and \(z_2\) such that \(|V(P)|\ge 6k+2r-2\). All the bounds are sharp and all results can be extended to 2k-factor-critical graphs.

令G为图，\（\ nu \）为G和k的阶数为正整数，使得\（k \ le（\ nu -2）/ 2 \）。然后ģ被说成是ķ -extendable如果它具有尺寸的匹配ķ和大小的每一个匹配ķ延伸至一个完美的匹配ģ。如果图G包含哈密顿循环，则为哈密顿。如果图G的任意两个顶点包含连接两个顶点的跨越路径，则它是哈密顿连通的。在本文中，我们讨论了k-可扩展非二分图\（\ kappa（G）\ ge 2k + r \）其中\（k \ ge 1 \）和\（r \ ge 0 \）。结果表明，如果\（\ nu \ le 6k + 2r \），则G为哈密顿量；如果\（\ nu> 6k + 2r \），则G具有最长的周期C，从而\（| V（C）| \ ge 6k + 2r \） ; 如果\（\ nu <6k + 2r \），则G是哈密顿连通的; 如果\（\ NU \ GE 6K + 2R \） ，然后为每个对顶点\（Z_1 \）和\（Z_2 \）的G ^，还有一个路径P的ģ接合\（Z_1 \）和\（z_2 \）这样\（| V（P）| \ ge 6k + 2r-2 \）。所有边界都是尖锐的，并且所有结果都可以扩展到2 k因子临界图。