论文标题
在较大的连通性或最小程度的挖掘中的长度限制与长度约束的脱节周期
Disjoint cycles with length constraints in digraphs of large connectivity or minimum degree
论文作者
论文摘要
Lichiardopol的一个猜想指出，每一个$ k \ ge 1 $都存在一个整数$ g（k）$，因此每个最低限度的挖掘物至少$ g（k）$包含$ k $ vertex-dischient ottertex-dischint otertex-dischient otertex-dischient otertex-dischint otertecles overtex-dischient otertecles otertex-dischient。由Lichiardopol的猜想所激发，我们研究了顶点 - 区连接的定向周期的存在，这些周期满足了较大的连通性或最低度较大度的挖掘过程中的长度约束。 Our main result is that for every $k \in \mathbb{N}$, there exists $s(k) \in \mathbb{N}$ such that every strongly $s(k)$-connected digraph contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths.相比之下，对于\ mathbb {n} $中的每一个$ k \，我们构造了一个强烈的$ k $连接的挖掘物，其中包含两个顶点或弧形 - 偶有长度的循环。无论是否存在$ g（3）$，这是一个空旷的问题。在这里，我们证明了整数$ k $的存在，以使每个最小和内部至少$ k $的挖掘都包含$ 3 $ tertex-disjoint的成对长度的定向循环。
A conjecture by Lichiardopol states that for every $k \ge 1$ there exists an integer $g(k)$ such that every digraph of minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. Motivated by Lichiardopol's conjecture, we study the existence of vertex-disjoint directed cycles satisfying length constraints in digraphs of large connectivity or large minimum degree. Our main result is that for every $k \in \mathbb{N}$, there exists $s(k) \in \mathbb{N}$ such that every strongly $s(k)$-connected digraph contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. In contrast, for every $k \in \mathbb{N}$ we construct a strongly $k$-connected digraph containing no two vertex- or arc-disjoint directed cycles of the same length. It is an open problem whether $g(3)$ exists. Here we prove the existence of an integer $K$ such that every digraph of minimum out- and in-degree at least $K$ contains $3$ vertex-disjoint directed cycles of pairwise distinct lengths.
