Some Properties of Graphs with Local Ore Condition

The author and N.K. Khachatrian proved that a connected graph \(G\) of order at least \(3\) is hamiltonian if for each vertex \(x\) the subgraph \(G_1(x)\) induced by \(x\) and its neighbors in \(G\) is an Ore graph.

We prove here that a graph \(G\) satisfying the above conditions is fully cycle extendible. Moreover, \(G\) is panconnected if and only if \(G\) is \(3\)-connected and \(G \neq K_n \lor \overline{K}_n\) for some \(n \geq 3\), where \(\lor\) is the join operation. The paper is concluded with two conjectures.