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.

💡 Research Summary

The paper addresses a long‑standing conjecture of Hoàng concerning degree‑sequence conditions for Hamiltonicity in t‑tough graphs. Chvátal’s classic theorem (1972) gives a degree‑sequence criterion for Hamilton cycles in arbitrary graphs: if for every index i < n/2 the inequality d_i ≤ i forces d_{n−i} ≥ n−i, then the graph is Hamiltonian. Hoàng (1995) extended this to 1‑tough graphs by replacing the right‑hand side with n−i+1, and proved the conjecture for t = 1, 2, 3. The case t ≥ 4 remained open.

The authors resolve the conjecture for all integers t ≥ 4 by introducing two new structural results that are of independent interest. The first, the “toughness closure lemma” (Theorem 5), shows that for any t‑tough graph G with t ≥ 4, if two non‑adjacent vertices x and y satisfy deg(x)+deg(y) ≥ n − t, then G is Hamiltonian if and only if the graph obtained by adding the edge xy is Hamiltonian. This is a natural generalisation of the classical Hamiltonian closure concept to the setting of toughness. Consequently, one may replace G by its t‑closure (the graph obtained by repeatedly adding edges between non‑adjacent pairs whose degree sum is at least n − t) without destroying the degree‑sequence condition.

The second key ingredient is a restricted cycle‑structure theorem (Theorem 6). It states that if a t‑tough graph G is non‑Hamiltonian but contains a vertex z whose removal leaves a Hamiltonian cycle C, then for any two distinct neighbours x, y of z that lie on C, the degree sum satisfies deg(x)+deg(y) < n − t. This result provides a powerful obstruction: any pair of neighbours of the “critical” vertex z with a large degree sum would immediately force Hamiltonicity, contradicting the assumption.

Armed with these tools, the authors conduct a detailed degree‑sequence analysis. They order the degrees d₁ ≤ d₂ ≤ … ≤ d_n and let k be the smallest index with d_k ≤ k. Because a t‑tough graph with t ≥ 4 has minimum degree at least 8, one obtains k ≥ 8 and d_{k−1}=d_k=k. For each integer α < n/2 they define the set U_α = {v_i : d_i ≥ n − α}. A series of claims establishes that (i) U_α forms a clique that is complete to all vertices of degree at least α − t, (ii) under certain conditions U_α becomes a universal clique (adjacent to every other vertex), and (iii) the size of any universal clique Ω is bounded above by k − 2.

Further combinatorial arguments (Claims 2.4–2.6) show that k must be at least n/2 − t, which forces the minimum degree δ(G) to exceed nt + 1 − 1. At this point the authors invoke a known result of Bauer et al. (2001) stating that any graph with δ(G) > nt + 1 − 1 is Hamiltonian. This yields a contradiction to the assumption that G is non‑Hamiltonian, thereby proving that any t‑tough graph satisfying Hoàng’s degree‑sequence condition is indeed Hamiltonian.

As a corollary (Corollary 4), the authors combine their main theorem with Hoàng’s earlier work on pancyclicity: a t‑tough graph (t ≥ 4) meeting the degree‑sequence condition is not only Hamiltonian but also pancyclic, because a bipartite graph of order at least three cannot be t‑tough for t > 1.

In summary, the paper settles Hoàng’s conjecture for all t ≥ 4, extending Chvátal’s and Hoàng’s earlier results to the full range of toughness parameters. The two novel structural theorems provide new techniques for handling degree‑sum conditions in tough graphs and are likely to find further applications in Hamiltonicity, pancyclicity, and related cycle‑structure problems. Future work may explore analogous closure lemmas for other graph invariants or extend the restricted cycle‑structure analysis to k‑factors, Hamiltonian paths, or to graphs with additional constraints such as bounded independence number or chromatic number.