  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 well-known theorem due to Erd\H{o}s 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 another applications, we obtain some results
which generalize previous theorems of Dirac, Erd\H{o}s-Gallai, Bondy, and
Fujisawa et al., present short proofs of the path case of
Loebl-Koml\'{o}s-S\'{o}s Conjecture which was verified by Bazgan et al. and of
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.
