arXiv 2104.063332021-04-13math.CO

超图的圈因子分解

Decomposing hypergraphs into cycle factors

Felix Joos, Marcus Kühn, Bjarne Schülke

备注：: 27 pages

摘要：

Rödl，Rucirobski和Szemerédi的一个著名结果保证了$k$ -一致超图$H$在$n$顶点上的（紧）Hamilton圈具有最小$(k-1)$ -度$δ_{k-1}(H)\geq (1/2+o(1))n$，从而将Dirac的结果从图推广到超图。对于图，我们知道的更多;每个在$n$顶点上具有$δ(G)\geq (1/2+o(1))n$的图都包含$(1-o(1))r$边不相交的Hamilton圈，其中$r$是最大整数，使得$G$包含一个生成$2r$ -正则子图，这显然是渐近最优的。费伯、克里维列维奇和苏达科夫在回答库恩、拉平斯卡斯和奥斯托斯提出的问题时证明了这一点。我们将这个结果推广到超图;每个$k$ -一致超图$H$的$n$顶点与$δ_{k-1}(H)\geq (1/2+o(1))n$包含$(1-o(1))r$边不相交（紧）的Hamilton圈，其中$r$是最大的整数，使得$H$包含一个生成子图，每个顶点属于$kr$边。特别是，这产生了一个问题的格洛克，库恩和奥斯托斯的渐近解。事实上，我们的主要结果适用于近似顶点正则$k$ -均匀超图的弱拟随机属性，并提供近似分解成周期因子没有太短的周期。

英文摘要：

A famous result by Rödl, Ruciński, and Szemerédi guarantees a (tight) Hamilton cycle in $k$-uniform hypergraphs $H$ on $n$ vertices with minimum $(k-1)$-degree $δ_{k-1}(H)\geq (1/2+o(1))n$, thereby extending Dirac's result from graphs to hypergraphs. For graphs, much more is known; each graph on $n$ vertices with $δ(G)\geq (1/2+o(1))n$ contains $(1-o(1))r$ edge-disjoint Hamilton cycles where $r$ is the largest integer such that $G$ contains a spanning $2r$-regular subgraph, which is clearly asymptotically optimal. This was proved by Ferber, Krivelevich, and Sudakov answering a question raised by Kühn, Lapinskas, and Osthus. We extend this result to hypergraphs; every $k$-uniform hypergraph $H$ on $n$ vertices with $δ_{k-1}(H)\geq (1/2+o(1))n$ contains $(1-o(1))r$ edge-disjoint (tight) Hamilton cycles where $r$ is the largest integer such that $H$ contains a spanning subgraph with each vertex belonging to $kr$ edges. In particular, this yields an asymptotic solution to a question of Glock, Kühn, and Osthus. In fact, our main result applies to approximately vertex-regular $k$-uniform hypergraphs with a weak quasirandom property and provides approximate decompositions into cycle factors without too short cycles.

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