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{\bf Ahmed Ainouche}
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{\bf Relaxations of Ore's Condition on Cycles}
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A simple, undirected $2$-connected graph $G$ of order $n$ belongs to
class ${\cal O}(n$,$\varphi)$, $\varphi\geq0$, if
$\sigma_{2}=n-\varphi.$ It is well known (Ore's theorem) that $G$ is
hamiltonian if $\varphi= 0$, in which case the $2$-connectedness
hypothesis is implied. In this paper we provide a method for studying
this class of graphs. As an application we give a full
characterization of graphs $G$ in ${\cal O}(n$,$\varphi)$,
$\varphi\leq3$, in terms of their dual hamiltonian closure.
\bye