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

TL;DR

The paper strengthens a degree-sequence condition for Hamiltonicity in $t$-tough graphs by proving the conjecture for all $t\ge4$ via a toughness-closure lemma and a Bondy–Chvátal type closure framework. The core approach is to assume non-Hamiltonicity and derive a $t$-closure structure that forces a large universal clique, which then triggers a Hamiltonicity criterion (Bauer et al.) to yield a contradiction. A key contribution is showing that the $t$-closure property holds precisely for $t\ge4$ and that the analogous lemma fails for $t=1$, demonstrated by a tight construction with $\tau(G)=1$. The results connect degree-sequence conditions to closure operations in tough graphs, advancing the understanding of when toughness and degree conditions guarantee Hamiltonicity.

Abstract

Generalizing Chvátal's classic 1972 result, Hoàng proposed in 1995 the following conjecture, which strengthens Chvátal's result in terms of toughness: Let $t\ge 1$ be a positive integer and $G$ be a $t$-tough graph on $n \ge 3$ vertices with degree sequence $d_1, d_2, \dots, d_n$ in non-increasing order. Suppose for each $i\in [1, \lfloor\frac{n-1}{2} \rfloor]$, if $d_i \le i \text{ and } d_{n-i+t} < n - i $ implies $d_j + d_{n-j+t} \ge n$ for all $j\in [i+1, \lfloor\frac{n-1}{2} \rfloor]$, then $G$ is Hamiltonian. Hoàng verified the conjecture for $t=1$. In this paper, we verfity the conjecture for all $t\ge 4$. Our proof relies on a toughness closure lemma for $t\ge 4$ that we previously established. Additionally, we show that the toughness closure lemma does not hold when $t=1$.

Figures (1)

• Figure 1: A depiction of $G$.

Theorems & Definitions (19)

• Theorem 1
• Conjecture 2
• Conjecture 3
• Theorem 4
• Theorem 5: Bondy and Chvátal bondychvatal1976
• Theorem 6: Shan and Tanyel shan2025degreesequence
• Theorem 7
• Theorem 8
• Theorem 9: Bauer et al. bauer1995long
• Theorem 10