Short Proofs of Some Fan-Type Results

For a graph \(G\), define \(\phi(G) = \min \{\max \{d(u), d(v)\} | d(u,v) = 2\}\) if \(G\) contains two vertices at distance 2, and \(\phi(G) = \infty\) otherwise. Fan proved that every 2-connected graph on \(n\) vertices with \(\phi(G) > \frac{1}{2}n\) is hamiltonian. Short proofs of this result and a number of analogues, some known, some new, are presented. Also, it is shown that if \(G\) is 2-connected, \(\phi(G) \geq \frac{1}{2}(n-i)\) and \(G – \{v \in V(G) | d(v) \geq \frac{1}{2} (n-i)\}\) has at least three components with more than \(i\) vertices, then \(G\) is hamiltonian (\(i \geq1\)).