arXiv 1807.050412018-07-13math.CO

图的笨拙填充

Clumsy packings of graphs

Maria Axenovich, Anika Kaufmann, Raphael Yuster

备注：: 14 pages, 3 figures

摘要：

设$G$和$H$是图。如果$P$是$G$中$H$的边不相交副本的集合，则我们说$P$是$G$的$H$ -packing。如果没有其它的$G$的$H$ -包装正确地包含$P$，则$H$ -包装$P$是最大的.最大基数的填充已被深入研究，最近有几个突破性的结果。在这里，我们考虑最小基数极大填充。如果一个$H$ -packing $P$是最大的或最小的，则它是笨拙的。令$cl(G,H)$为$G$的笨拙$H$ -包装的大小。我们提供了非平凡的界限$cl(G,H)$，并在许多情况下渐近确定$cl(G,H)$的一些通用类的图形$G$，如$K_n$（完全图），$Q_n$（立方体图），以及正方形，三角形和六边形网格。对于每个固定的非空图$H$，我们渐近地确定了$cl(K_n,H)$。特别地，我们证明了$$ cl(K_n, H) = \frac{\binom{n}{2}- ex(n,H)}{|E(H)|}+o(ex(n,H)),$$，其中$ex(n,H)$是$H$的极值数。一个相关的自然参数是$cov(G,H)$，它是$G$中$H$的最小拷贝数（不一定是边不相交的），从$G$中去除它会产生一个无$H$的图。虽然显然是$cov(G,H) \le cl(G,H)$，但我们对$cl(G,H)$的所有下限也适用于$cov(G,H)$。

英文摘要：

Let $G$ and $H$ be graphs. We say that $P$ is an $H$-packing of $G$ if $P$ is a set of edge-disjoint copies of $H$ in $G$. An $H$-packing $P$ is maximal if there is no other $H$-packing of $G$ that properly contains $P$. Packings of maximum cardinality have been studied intensively, with several recent breakthrough results. Here, we consider minimum cardinality maximal packings. An $H$-packing $P$ is clumsy if it is maximal of minimum size. Let $cl(G,H)$ be the size of a clumsy $H$-packing of $G$. We provide nontrivial bounds for $cl(G,H)$, and in many cases asymptotically determine $cl(G,H)$ for some generic classes of graphs $G$ such as $K_n$ (the complete graph), $Q_n$ (the cube graph), as well as square, triangular, and hexagonal grids. We asymptotically determine $cl(K_n,H)$ for every fixed non-empty graph $H$. In particular, we prove that $$ cl(K_n, H) = \frac{\binom{n}{2}- ex(n,H)}{|E(H)|}+o(ex(n,H)),$$ where $ex(n,H)$ is the extremal number of $H$. A related natural parameter is $cov(G,H)$, that is the smallest number of copies of $H$ in $G$ (not necessarily edge-disjoint) whose removal from $G$ results in an $H$-free graph. While clearly $cov(G,H) \le cl(G,H)$, all of our lower bounds for $cl(G,H)$ apply to $cov(G,H)$ as well.

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。

摘要：

英文摘要：

相关论文