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超图边集的几乎正则圈划分

Partitioning The Edge Set of a Hypergraph Into Almost Regular Cycles

Amin Bahmanian, Sadegheh Haghshenas

备注：
17 pages
发表信息：
Journal of Combinatorial Designs, Volume 26, Issue 10, October 2018, Pages 465-479
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摘要：

在超图中，长度为$t$的圈是由不同的顶点$v_i$和不同的边$e_i$构成的交替序列$v_1,e_1,v_2\dots,v_t,e_t$，使得$\{v_i,v_{i+1}\}\subseteq e_i$（具有$v_{t+1}:=v_1$）。设$λK_n^h$是$λ$ -fold $n$ -顶点完全$h$ -图.设$\mathcal G=(V,E)$是一个超图，其所有边的大小至少为$h$和$2\leq c_1\leq \dots\leq c_k\leq |V|$。为了将$\mathcal G$的边集划分为指定长度的圈$c_1, \dots, c_k$，一个明显的必要条件是$\sum_{i=1}^k c_i=|E|$。我们证明了这个条件在下列情况下是充分的：（i）$h\geq \max\{c_k, \lceil n/2 \rceil+1\}$ ;（ii）$\mathcal G=λK_n^h$，$h\geq \lceil n/2 \rceil+2$ ;（iii）$\mathcal G=K_n^h$，$c_1= \dots=c_k:=c$，$c|n(n-1), n\geq 85$ .在（ii）中，我们保证每个周期几乎是正则的。在（iii）中，我们还解决了$K_n^h$的边缘的“小”子集$L$被移除的情况。

英文摘要：

A cycle of length $t$ in a hypergraph is an alternating sequence $v_1,e_1,v_2\dots,v_t,e_t$ of distinct vertices $v_i$ and distinct edges $e_i$ so that $\{v_i,v_{i+1}\}\subseteq e_i$ (with $v_{t+1}:=v_1$). Let $λK_n^h$ be the $λ$-fold $n$-vertex complete $h$-graph. Let $\mathcal G=(V,E)$ be a hypergraph all of whose edges are of size at least $h$, and $2\leq c_1\leq \dots\leq c_k\leq |V|$. In order to partition the edge set of $\mathcal G$ into cycles of specified lengths $c_1, \dots, c_k$, an obvious necessary condition is that $\sum_{i=1}^k c_i=|E|$. We show that this condition is sufficient in the following cases: (i) $h\geq \max\{c_k, \lceil n/2 \rceil+1\}$; (ii) $\mathcal G=λK_n^h$, $h\geq \lceil n/2 \rceil+2$; (iii) $\mathcal G=K_n^h$, $c_1= \dots=c_k:=c$, $c|n(n-1), n\geq 85$. In (ii), we guarantee that each cycle is almost regular. In (iii), we also solve the case where a "small" subset $L$ of edges of $K_n^h$ is removed.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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