Extremal subgraphs of random graphs
Extremal subgraphs of random graphs
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随机图的极值子图
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
A. Steger
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文献类型：
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作者：
G. Brightwell;K. Panagiotou;A. Steger
我们证明了存在一个常数c > 0，使得当p ≥ n-c时，当n趋于无穷大时概率趋于1，随机图Gn，p的每个最大无三角子图都是二部图。这回答了巴拜、Simonovits和Spencer的问题（巴拜等人，J Graph Theory 14（1990）599-622）。证明是基于一个工具的独立利益：例如，我们表明，几乎所有的图与M边，其中M n和M ≤ $（矩阵{ncr 2 cr }）$ /2的最大割是“几乎唯一的”。更准确地说，给定Gn，M的最大切割C，我们可以通过移动最多documentclass{article}usepackage{mathrsfs，amsmath，amssymb}pagestyle{empty}来获得所有最大切割。登录{文档}egin{align*}mathcal{O}（sqrt{n^3/M}）end{align*}end{document}顶点之间的部分C。© 2012 Wiley Periodicals，Inc.随机结构算法，2012
We prove that there is a constant c > 0, such that whenever p ≥ n‐c, with probability tending to 1 when n goes to infinity, every maximum triangle‐free subgraph of the random graph Gn,p is bipartite. This answers a question of Babai, Simonovits and Spencer (Babai et al., J Graph Theory 14 (1990) 599–622). The proof is based on a tool of independent interest: we show, for instance, that the maximum cut of almost all graphs with M edges, where M ≫ n and M ≤ $(matrix{ n cr 2 cr } )$ /2, is “nearly unique”. More precisely, given a maximum cut C of Gn,M, we can obtain all maximum cuts by moving at most documentclass{article}usepackage{mathrsfs, amsmath, amssymb}pagestyle{empty}egin{document}egin{align*}mathcal{O}(sqrt{n^3/M})end{align*}end{document} vertices between the parts of C. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012