Degree sequence condition for pancyclicity in tough graphs

Source abstract

Let $t \ge 1$ be an integer, and let $G$ be a $t$-tough $n$-vertex graph with degree sequence $d_1, d_2, \ldots, d_n$ in non-decreasing order. In 1995, Hoàng conjectured that if $G$ is Hamiltonian and, for every integer $i$ satisfying $t\le i<n/2$, $d_i\le i$, and $d_{n-i+t}<n-i$, one has $d_j + d_{n-j+t} \ge n$ for all $j$ with $i < j < \frac{n}{2}$, then $G$ is pancyclic or bipartite. In this paper, we disprove the conjecture for $t = 1$ and confirm it for all $t \ge 7$.