A strengthening of a degree sequence condition for Hamiltonicity in tough graphs

Generalizing Chv\'atal's classic 1972 result, Ho\`ang proposed in 1995 the following conjecture, which strengthens Chv\'atal's result in terms of toughness: Let t≥ 1 be a positive integer and G be a t-tough graph on n ≥ 3 vertices with degree sequence d₁, d₂, , dₙ in non-increasing order. Suppose for each i∈ [1, -1/2 ], if dᵢ ≤ i and dₙ₋ᵢ₊ₜ < n - i implies dⱼ + dₙ₋ⱼ₊ₜ ≥ n for all j∈ [i+1, -1/2 ], then G is Hamiltonian. Ho\`ang verified the conjecture for $t=1$. In this paper, we verfity the conjecture for all t≥ 4. Our proof relies on a toughness closure lemma for t≥ 4 that we previously established. Additionally, we show that the toughness closure lemma does not hold when $t=1$.