Abstract: This talk has two focuses, both concerning Hamiltonicity conditions in tough graphs. First, we present a $t$-closure lemma that generalizes a closure lemma of Bondy and Chvátal from 1976. In 1995, Hoàng generalized Chvátal's degree sequence condition for Hamiltonicity in 1-tough graphs and  posed two $t$-tough analogues  for any positive integer $t \ge 1$. Hoàng confirmed his conjectures respectively  for $t \le 3$ and $t=1$. We apply our $t$-closure lemma to confirm both conjectures for all $t \ge 4$.

Second, we present that all 71-tough $(2P_2 \cup P_1)$-free graphs of order at least three are Hamiltonian. This research is inspired by Chvátal's conjecture that there exists a constant $t_0$ such that all $t_0$-tough graphs of order at least three are Hamiltonian. This conjecture is still open, but work has been done to find such a $t_0$ for certain graph classes. With our result, the conjecture is confirmed for all $F$-free graphs where $F$ is any five-vertex linear forest excepting $P_5$.