A note on a degree sum condition for long cycles in graphs
Abstract
We conjecture that a 2connected graph $G$ of order $n$, in which $d(x)+d(y)\geq nk$ for every pair of nonadjacent vertices $x$ and $y$, contains a cycle of length $nk$ ($k<n/2$), unless $G$ is bipartite and $nk$ is odd. This generalizes to long cycles a wellknown degree sum condition for hamiltonicity of Ore. The conjecture is shown to hold for $k=1$.
 Publication:

arXiv eprints
 Pub Date:
 November 2007
 arXiv:
 arXiv:0711.4394
 Bibcode:
 2007arXiv0711.4394A
 Keywords:

 Mathematics  Combinatorics;
 05C38
 EPrint:
 5 pages