A note on a degree sum condition for long cycles in graphs

We conjecture that a 2-connected graph $G$ of order $n$, in which $d(x)+d(y)\ge n-k$ for every pair of non-adjacent vertices $x$ and $y$, contains a cycle of length $n-k$ ($k<n\mathrm{/}2$), unless $G$ is bipartite and $n-k$ is odd. This generalizes to long cycles a well-known degree sum condition for hamiltonicity of Ore. The conjecture is shown to hold for $k=1$.