A strengthening of Erd?s-Gallai Theorem and proof of Woodall's conjecture

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