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arXiv 1101.3099math.COcs.DMmath.PR
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随机图中Hamilton圈的Hamilton性和最优填充的恢复性

On the resilience of Hamiltonicity and optimal packing of Hamilton cycles in random graphs

Sonny Ben-Shimon, Michael Krivelevich, Benny Sudakov

备注：
20 pages
发表信息：
SIAM Journal on Discrete Mathematics, 25(3):1176--1193, 2011
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摘要：

令$\bk=(k_1,...,k_n)$是$n$整数序列。对于一个单调递增图性质$\mP$，我们说一个基图$G=([n],E)$是关于$\mP$的\emph{$\bk$-resilient}，如果对于每个子图$H\subseteq G$，使得对于每个$1\leq i\leq n$，图$G-H$具有$\mP$，$d_H(i)\leq k_i$。这个概念自然地扩展了最近由Sudakov和Vu提出的图的\emph{local resilience}的思想。在本文中，我们研究了$\bk$ -弹性的一个典型的图从$\GNP$相对于汉密尔顿性质，我们让$p$范围内的所有值的基础图预计是汉密尔顿。特别是，我们证明了，对于每个$ε>0$和$p\geq\frac{\ln n+\ln\ln n +ω(1)}{n}$，如果一个图是从$\GNP$采样，然后以高概率删除从每个顶点的“小”度的所有事件的边缘，但两个和从任何其他顶点至多一个$(\frac{1}{3}-ε)$ -分数的事件的边缘将导致在一个哈密尔顿图。考虑到这种广义的方法的概念弹性允许建立几个推论，提高最知名的界限的哈密顿相关的问题。这意味着，对于每个正的$ε>0$和足够大的$K$值，如果$p>\frac{K\ln n}{n}$，则$\GNP$相对于哈密顿量的局部弹性很可能至少是$(1-ε)np/3$，改进了$p$的该范围的先前界。另一个含义是随机图中边不交Hamilton圈的最优填充的结果。我们证明，如果$p\leq\frac{1.02\ln n}{n}$然后以高概率从$\GNP$采样图$G$包含$\lfloor\frac{δ(G)}{2}\rfloor$边不相交的汉密尔顿周期，扩展以前的范围$p$，这是已知的。

英文摘要：

Let $\bk=(k_1,...,k_n)$ be a sequence of $n$ integers. For an increasing monotone graph property $\mP$ we say that a base graph $G=([n],E)$ is \emph{$\bk$-resilient} with respect to $\mP$ if for every subgraph $H\subseteq G$ such that $d_H(i)\leq k_i$ for every $1\leq i\leq n$ the graph $G-H$ possesses $\mP$. This notion naturally extends the idea of the \emph{local resilience} of graphs recently initiated by Sudakov and Vu. In this paper we study the $\bk$-resilience of a typical graph from $\GNP$ with respect to the Hamiltonicity property where we let $p$ range over all values for which the base graph is expected to be Hamiltonian. In particular, we prove that for every $ε>0$ and $p\geq\frac{\ln n+\ln\ln n +ω(1)}{n}$ if a graph is sampled from $\GNP$ then with high probability removing from each vertex of "small" degree all incident edges but two and from any other vertex at most a $(\frac{1}{3}-ε)$-fraction of the incident edges will result in a Hamiltonian graph. Considering this generalized approach to the notion of resilience allows to establish several corollaries which improve on the best known bounds of Hamiltonicity related questions. It implies that for every positive $ε>0$ and large enough values of $K$, if $p>\frac{K\ln n}{n}$ then with high probability the local resilience of $\GNP$ with respect to being Hamiltonian is at least $(1-ε)np/3$, improving on the previous bound for this range of $p$. Another implication is a result on optimal packing of edge disjoint Hamilton cycles in a random graph. We prove that if $p\leq\frac{1.02\ln n}{n}$ then with high probability a graph $G$ sampled from $\GNP$ contains $\lfloor\frac{δ(G)}{2}\rfloor$ edge disjoint Hamilton cycles, extending the previous range of $p$ for which this was known to hold.

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