For a 2-connected graph G on n vertices and two vertices $x,y\in V(G)$, we prove that there is an $(x,y)$-path of length at least k, if there are at least $\frac{n-1}{2}$ vertices in $V(G)\backslash \{x,y\}$ of degree at least k. This strengthens a celebrated theorem due to Erdős and Gallai in 1959. As the first application of this result, we show that a 2-connected graph with n vertices contains a cycle of length at least 2k, if it has at least $\frac{n}{2}+k$ vertices of degree at least k. This confirms a 1975 conjecture made by Woodall. As other applications, we obtain some results which generalize previous theorems of Dirac, Erdős-Gallai, Bondy, and Fujisawa et al., present short proofs of the path case of Loebl-Komlós-Sós Conjecture which was verified by Bazgan et al. and a conjecture of Bondy on longest cycles (for large graphs) which was confirmed by Fraisse and Fournier, and make progress on a conjecture of Bermond.

对于2个连通图G的n个顶点和两个顶点$X，ÿ\in V（G）$，我们证明有一个 $（X，ÿ）$-长度至少为k的路径，如果至少存在$\frac{ñ-\mathrm{1个}}{2}$ 顶点 $V（G）\backslash \{X，ÿ\}$的度数至少为k。这加强了1959年由Erdős和Gallai提出的著名定理。作为该结果的首次应用，我们证明了2个具有n个顶点的连通图包含一个长度至少为2 k的循环，如果它具有至少一个$\frac{ñ}{2}+ķ$顶点度至少为k。这证实了伍德奥尔（Woodall）在1975年提出的猜想。作为其他应用，我们获得了一些结果，这些结果推广了Dirac，Erdős-Gallai，Bondy和Fujisawa等人的先前定理，为Loebl-Komlós-Sós猜想的路径情形提供了简短证明，并得到了Bazgan等人的验证。Fraisse和Fournier证实了最长周期的邦迪猜想（对于大图），而伯蒙德的猜想也在不断发展。