Divergence Condition for Pancyclicity

Let \(G\) be a connected graph and let \(u\) and \(v\) be two vertices of \(G\) such that \(d_G(u, v) = 2\). We define their divergence to be: \(\alpha^*(u, v) = \max_{w} \{ |S| \mid for each w \in N_G(u) \cap N_G(v), S is a maximum independent set in N_G(w) containing u \text{ and } v \}.\) It is proved that if for each pair of vertices \(u\) and \(v\) of \(G\) such that \(d_G(u, v) = 2\), \(|N_G(u) \cap N_G(v)| \geq \alpha^*(u, v)\) and if \(\nu(G) \geq 3\), then \(G\) is pancyclic unless \(G\) is \(K_{{\nu}/{2},{\nu}/{2}}\). Several previously known sufficient conditions for pancyclicity follow as corollaries.