A generalization of Bondy's and Fan's sufficient conditions for Hamiltonian graphs (Q1357748)

Chen et al. (1994) showed that for a \(k\)-connected graph \(G\) on \(n \geq 3\) vertices, if the maximum degree in any ``essential'' (i.e., containing two vertices a distance 2 apart) independent set of \(k\) vertices is at least \(n/2\), then \(G\) is Hamiltonian. The authors of this paper generalize the above result (and previous results of Fan and Bondy) by showing that for such \(G\), if the previous degree condition holds for any essential independent set on \(k+1\) vertices, then either \(G\) is Hamiltonian or is one of three types of graphs.