An Ore-type theorem for $[3]$-graphs

TL;DR

The paper proves an Ore-type condition for Hamiltonian Berge cycles in [3]-graphs: there exists a constant $d_0$ such that if every nonadjacent pair $u,v$ satisfies $d_{\,\partial \mathcal{H}}(u)+d_{\,\partial \mathcal{H}}(v) \ge n+d_0$, then $\mathcal{H}$ contains a Hamiltonian Berge cycle; the authors establish this with $d_0=65$ and conjecture that the bound $d_0=1$ would suffice. The method adapts classical graph techniques to hypergraphs via the 2-shadow and Berge cycles, introducing usable sets for outside vertices and a bridge-counting argument around a maximal Berge cycle. A key contribution is the detailed structural lemmas (including a main lemma with parts (a)–(e)) that enable a contradiction by counting bridges against the cycle’s edge budget. The work connects to covering hypergraphs (where the result is immediate) and opens avenues for generalization to higher uniformities and varying $n_0$, offering a framework for Ore-type results beyond graphs.

Abstract

Ore's Theorem states that if $G$ is an $n$-vertex graph and every pair of non-adjacent vertices has degree sum at least $n$, then $G$ is Hamiltonian. A $[3]$-graph is a hypergraph in which every edge contains at most $3$ vertices. In this paper, we prove an Ore-type result on the existence of Hamiltonian Berge cycles in $[3]$-graph $\cH$, based on the degree sum of every pair of non-adjacent vertices in the $2$-shadow graph $\partial \cH$ of $\cH$. Namely, we prove that there exists a constant $d_0$ such that for all $n \geq 6$, if a $[3]$-graph $\cH$ on $n$ vertices satisfies that every pair $u,v \in V(\cH)$ of non-adjacent vertices has degree sum $d_{\partial \cH}(u) + d_{\partial \cH}(v) \geq n+d_0$, then $\cH$ contains a Hamiltonian Berge cycle. Moreover, we conjecture that $d_0=1$ suffices.

Figures (5)

• Figure 1: $v_i, v_j \in U_u$ and $f \notin \{e_i, e_j\}$.
• Figure 2: (Left) Bridges produced by $\{v_i,v_j\}$ where $v_i \to v_k \to v_j$. (Right) The same bridge $e=\{v_i,v_{k+1},v_{k+2}\}$ is produced by all of the pairs $\{v_i,v_{j_k}\}$ where $k=1,2,3$.
• Figure 3: $v_i, v_j \in U_u$, $f_i \notin \{e_{k-1},e_i\}$, and $f_j \notin \{e_{k+1},e_j\}$.
• Figure 4: $v_{i+1},v_{j+1} \in X \cap S'({\cal C})$.
• Figure 5: $v_{i+1}, v_{j+1} \in X$, $f_{i+1} \notin \{e_{k-1},e_{i+1}\}$, and $f_{j+1} \notin \{e_{k+1},e_{j+1}\}$.

Theorems & Definitions (31)

• Theorem 1: Dirac's Theorem D
• Theorem 2: Ore's Theorem Ore
• Theorem 3: Coulson-Perarnau CP
• Theorem 4: Coulson-Perarnau CP
• Theorem 5: Kostochka, Luo, McCourt KLM
• Conjecture 7
• Theorem 8
• Theorem 9: Lu-WangLW
• Claim 10
• proof