Degree sequence condition for Hamiltonicity in tough graphs

Generalizing both Dirac’s condition and Ore’s condition for Hamilton cycles, Chvátal in 1972 established a degree sequence condition for the existence of a Hamilton cycle in a graph. Hoàng in 1995 generalized Chvátal’s degree sequence condition for 1-tough graphs and conjectured a $t$-tough analogue for any positive integer $t\ge 1$. Hoàng in the same paper verified his conjecture for $t\le 3$ and recently Hoàng and Robin verified the conjecture for $t=4$. In this paper, we confirm the conjecture for all $t\ge 4$. The proof depends on two newly established results on cycle structures in tough graphs, which hold independent interest.