On a conjecture of Grünbaum on longest cycles

Grünbaum conjectured that for any integer k≥2, there exists no n-vertex graph G of circumference n−k in which the removal of any k vertices from G yields a hamiltonian graph. We show that for any positive integers c and k there is an infinite family of c-connected graphs of circumference k less than their order, in which the limit (as the graphs’ order tends to infinity) of the ratio between the number of k-vertex sets whose removal yields a hamiltonian graph and the number of all k-vertex sets is 1. Motivated by a question of Katona, Kostochka, Pach, and Stechkin, it is proven that there exists an infinite family of non-hamiltonian graphs of increasing diameter d in which the removal of any two vertices at distance 1 or any distance at least (d+6)/2 yields a hamiltonian graph.