最小度条件下Berge哈密顿圈的稳定性定理
A stability theorem for Berge Hamiltonian cycles under a minimum degree condition
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中文总结 AI 辅助
该研究针对r-均匀超图，在最小度条件下，利用Salia的Pósa型定理得出不含Berge哈密顿圈的超边数极值上界，并证明近极值实例属于两类非哈密顿构造，为Berge哈密顿圈的稳定性提供了理论支撑。
中文摘要 AI 辅助
本文研究r-均匀超图中Berge哈密顿圈的极值问题与稳定性问题，在最小度条件下展开分析。定义函数g_r(n,t)=C(n-t,r)+t*C(t,r-1)，令t=t(k)为满足C(t-1,r-1)<k≤C(t,r-1)的唯一整数。利用Salia的Pósa型尖锐度序列定理，证明了n顶点r-均匀超图中，最小度至少为k且不含Berge哈密顿圈的超边数的极值上界；还在g_r(n,t)的第一个极小值点前的稠密范围内证明了稳定性定理：每个近极值实例都包含在两种自然的非哈密顿构造之一中。
英文摘要
In this paper, we study extremal and stability problems for Berge Hamiltonian cycles in $r$-uniform hypergraphs under a minimum degree condition. Let $ g_r(n,t)=\binom{n-t}{r}+t\binom{t}{r-1}$, and let $t=t(k)$ be the unique integer satisfying $\binom{t-1}{r-1}<k\le \binom{t}{r-1}$. Using a sharp Pósa-type degree sequence theorem of Salia, we prove an extremal upper bound on the number of hyperedges in an $n$-vertex $r$-uniform hypergraph with minimum degree at least $k$ and with no Berge Hamiltonian cycle. We also prove a stability theorem in the dense range before the first minimizer of $g_r(n,t)$: every near-extremal example is contained in one of two natural non-Hamiltonian constructions.
