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Degree sequence condition for Hamiltonicity in tough graphs

Songling Shan, Arthur Tanyel

TL;DR

This work settles Hoàng’s degree-sequence conjecture for all t≥4 by developing a toughness-closure framework. It proves a t-closure based reduction and introduces two key cycle-structure results that enforce Hamiltonicity under a Chvátal-style degree-sequence condition: if for all i< n/2, d_i ≤ i implies d_{n−i+t} ≥ n−i, then any t-tough graph is Hamiltonian. The authors derive a universal clique through careful degree-sequence analysis (and leverage Bauer et al.) to conclude Hamiltonicity, and they establish a closure lemma and a sum-of-successors lemma as foundational tools. Collectively, the results extend classical Dirac/Ore-type conditions to tough graphs, yielding a unified criterion for Hamiltonicity (and relatedly, pancyclicity) in this broader setting.

Abstract

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.

Degree sequence condition for Hamiltonicity in tough graphs

TL;DR

This work settles Hoàng’s degree-sequence conjecture for all t≥4 by developing a toughness-closure framework. It proves a t-closure based reduction and introduces two key cycle-structure results that enforce Hamiltonicity under a Chvátal-style degree-sequence condition: if for all i< n/2, d_i ≤ i implies d_{n−i+t} ≥ n−i, then any t-tough graph is Hamiltonian. The authors derive a universal clique through careful degree-sequence analysis (and leverage Bauer et al.) to conclude Hamiltonicity, and they establish a closure lemma and a sum-of-successors lemma as foundational tools. Collectively, the results extend classical Dirac/Ore-type conditions to tough graphs, yielding a unified criterion for Hamiltonicity (and relatedly, pancyclicity) in this broader setting.

Abstract

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 -tough analogue for any positive integer . Hoàng in the same paper verified his conjecture for and recently Hoàng and Robin verified the conjecture for . In this paper, we confirm the conjecture for all . The proof depends on two newly established results on cycle structures in tough graphs, which hold independent interest.
Paper Structure (4 sections, 7 theorems, 14 equations, 3 figures)

This paper contains 4 sections, 7 theorems, 14 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. Proof of Theorem \ref{['thm:t-closure']}
  4. Proof of Theorem \ref{['thm:sum-of-successors']}

Key Result

Theorem 1

Let $G$ be a graph with degree sequence $d_1, d_2, \dots, d_n$, where $n\ge 3$ is an integer. If for all $i < \frac{n}{2}$, $d_i \le i$ implies $d_{n-i} \ge n-i$, then $G$ is Hamiltonian.

Figures (3)

  • Figure 1: Construction of $C$ when $i<j$, drawn in red. The green arrows indicate the orientation of the corresponding segments of $P$ on $\overset{\rightharpoonup }{C}$.
  • Figure 2: A Hamilton $(x,y)$-path $P$ of $G$ constructed from $C$, where the first green arrow indicates that the direction from $x$ to $y^-$ on $P$ consists with the clockwise direction of $C$, and the second green arrow indicates that the direction from $x^-$ to $y$ on $P$ is opposite to the clockwise direction of $C$.
  • Figure 3: Hamilton cycles in $G$ under the assumption that there exists $w\in (Z\setminus Y) \cup R$ such that there are three distinct vertices $v_1, v_2, v_3\in N(U)\setminus N(U,P)$ for which $g(v_1)=g(v_2)=g(v_3)=w$.

Theorems & Definitions (28)

  • Theorem 1
  • Conjecture 2
  • Theorem 3
  • Corollary 4
  • Theorem 6
  • Theorem 7
  • Claim 2.1
  • Claim 2.2
  • Claim 2.3
  • Claim 2.4
  • ...and 18 more