[Paper Review] A Dirac-type theorem for uniform hypergraphs

Yue Ma, Xinmin Hou+1 more|arXiv (Cornell University)|Apr 10, 2020

Limits and Structures in Graph Theory19 references4 citations

TL;DR

This paper establishes a Dirac-type minimum degree condition for $r$-uniform hypergraphs to guarantee the existence of long Berge paths and cycles. It proves that if a connected $n$-vertex $r$-graph has minimum degree exceeding $\binom{k}{r-1}$, then it contains a Berge path of length at least $2k+1$, with tight extremal constructions characterized for $k > r \geq 4$ and $n > 2k+1$. The result extends Dirac's theorem to hypergraphs and improves bounds for Berge Hamiltonian cycles.

ABSTRACT

Dirac (1952) proved that every connected graph of order $n>2k+1$ with minimum degree more than $k$ contains a path of length at least $2k+1$. Erdős and Gallai (1959) showed that every $n$-vertex graph $G$ with average degree more than $k-1$ contains a path of length $k$. The hypergraph extension of the Erdős-Gallai Theorem have been given by Győri, Katona, Lemons~(2016) and Davoodi et al.~(2018). Füredi, Kostochka, and Luo (2019) gave a connected version of the Erdős-Gallai Theorem for hypergraphs. In this paper, we give a hypergraph extension of the Dirac's Theorem: Given positive integers $n,k$ and $r$, let $H$ be a connected $n$-vertex $r$-graph with no Berge path of length $2k+1$. We show that (1) If $k> r\ge 4$ and $n>2k+1$, then $δ_1(H)\le\binom{k}{r-1}$. Furthermore, the equality holds if and only if $S'_r(n,k)\subseteq H\subseteq S_r(n,k)$ or $H\cong S(sK_{k+1}^{(r)},1)$; (2) If $k\ge r\ge 2$ and $n>2k(r-1)$, then $δ_1(H)\le \binom{k}{r-1}$. The result is also a Dirac-type version of the result of Füredi, Kostochka, and Luo. As an application of (1), we give a better lower bound of the minimum degree than the ones in the Dirac-type results for Berge Hamiltonian cycle given by Bermond et al.~(1976) and Clemens et al. (2016), respectively.

Motivation & Objective

• To extend Dirac's classical path existence theorem from graphs to $r$-uniform hypergraphs.
• To determine the minimum degree threshold ensuring the existence of a Berge path of length $2k+1$ in connected $r$-graphs.
• To characterize the extremal hypergraphs achieving the threshold when $k > r \geq 4$ and $n > 2k+1$.
• To apply the result to improve lower bounds for Berge Hamiltonian cycle conditions in hypergraphs.

Proposed method

• Define Berge paths and cycles in $r$-uniform hypergraphs using vertex-edge incidence relations.
• Introduce extremal hypergraph constructions $S_r(n,k)$, $S'_r(n,k)$, and $S(sK_{k+1}^{(r)},1)$ to analyze extremal cases.
• Use degree arguments and structural analysis on longest Berge paths to bound minimum degree.
• Apply convexity and combinatorial inequalities to bound the degree of endpoints of longest Berge paths.
• Prove a key lemma showing that high minimum degree forces a Berge Hamiltonian cycle when $n = t+1$ and $\delta_1 > \binom{\lfloor t/2\rfloor}{r-1} + \lceil t/2 \rceil$.
• Use contradiction and vertex partitioning to rule out long Berge paths or cycles under low degree assumptions.

Experimental results

Research questions

• RQ1 What is the minimum degree threshold in $r$-uniform hypergraphs that guarantees a Berge path of length $2k+1$?
• RQ2 For which $r$-graphs is the bound $\delta_1(H) \leq \binom{k}{r-1}$ tight, and what are the extremal structures?
• RQ3 Can the Dirac-type condition for Berge Hamiltonian cycles be improved using the new path existence threshold?
• RQ4 How does the connectedness of the hypergraph affect the extremal degree threshold for long Berge paths?
• RQ5 What is the role of the constructions $S_r(n,k)$, $S'_r(n,k)$, and $S(sK_{k+1}^{(r)},1)$ in characterizing extremal hypergraphs?

Key findings

• For $k > r \geq 4$ and $n > 2k+1$, if $H$ is a connected $n$-vertex $r$-graph with no Berge path of length $2k+1$, then $\delta_1(H) \leq \binom{k}{r-1}$, with equality if and only if $S'_r(n,k) \subseteq H \subseteq S_r(n,k)$ or $H \cong S(sK_{k+1}^{(r)},1)$.
• For $k \geq r \geq 2$ and $n > 2k(r-1)$, the same bound $\delta_1(H) \leq \binom{k}{r-1}$ holds for connected $r$-graphs with no Berge path of length $2k+1$.
• The extremal constructions $S_r(n,k)$, $S'_r(n,k)$, and $S(sK_{k+1}^{(r)},1)$ all have minimum degree exactly $\binom{k}{r-1}$ and contain no Berge path longer than $2k$.
• A new lemma shows that if $\delta_1(H) > \binom{\lfloor t/2\rfloor}{r-1} + \lceil t/2 \rceil$ and $H$ is connected with longest Berge path length $t$, then $n = t+1$ and $H$ contains a Berge Hamiltonian cycle.
• The result improves previous lower bounds for Berge Hamiltonian cycles: when $\delta_1(H) > \binom{k}{r-1} + k+1$, $H$ contains a Berge Hamiltonian cycle for $n = 2k+2$ or $n = 2k+3$, depending on parity.
• The paper establishes a hypergraph extension of Dirac’s theorem, providing the first Dirac-type result for Berge paths and cycles in uniform hypergraphs.