论文标题
DIRAC定理的超图类似物在2个连接图中长期循环
A hypergraph analog of Dirac's Theorem for long cycles in 2-connected graphs
论文作者
论文摘要
狄拉克证明，每个$ n $ vertex $ 2 $ 2 $连接的图形至少$ k $包含一个长度周期至少$ \ min \ {2k，n \} $。我们考虑了此结果的超图版本。 HyperGraph中的BERGE周期是不同顶点的交替序列，并且边缘$ v_1，e_2，v_2，\ ldots，e_c，e_c，v_1 $，以至于$ \ {v_i，v_ {i+1} \ \} \ subSeteq e_i $ for Indece for Indece for Indece a Modulo $ c $ $ c $）。我们证明，对于$ n \ geq k \ geq r + 2 \ geq 5 $，每$ 2 $ - 连接的$ r $ r $ rubiform $ n $ n $ n $ n $ n $ vertex Hypergraph，至少至少$ {k-1 \ select r-1} + 1 $具有长度的berge clece of lengues of Length Cypeciptive $ \ \最min \ \ \ \ {2k {2k {2k，n \ n \ n \ n \ n \ n \ n \ n \ n \ n \ n。对于所有$ k \ geq r+2 \ geq 5 $，限制都是确切的。
Dirac proved that each $n$-vertex $2$-connected graph with minimum degree at least $k$ contains a cycle of length at least $\min\{2k, n\}$. We consider a hypergraph version of this result. A Berge cycle in a hypergraph is an alternating sequence of distinct vertices and edges $v_1,e_2,v_2, \ldots, e_c, v_1$ such that $\{v_i,v_{i+1}\} \subseteq e_i$ for all $i$ (with indices taken modulo $c$). We prove that for $n \geq k \geq r+2 \geq 5$, every $2$-connected $r$-uniform $n$-vertex hypergraph with minimum degree at least ${k-1 \choose r-1} + 1$ has a Berge cycle of length at least $\min\{2k, n\}$. The bound is exact for all $k\geq r+2\geq 5$.
