arXiv 1505.030722015-05math.COcs.DM

完全子图

Full subgraphs

Victor Falgas-Ravry, Klas Markström, Jacques Verstraëte

备注：: 18 pages

摘要：

设$G=(V,E)$是$p$在$n$顶点上的密度图。根据Erdés，Schluczak和Spencer的观点，如果$G$的$m$ -顶点子图$H$的最小度至少为$p(m - 1)$，则称$H$为{ \em full}.令$f(G)$表示$G$的最大满子图的阶。如果$p\binom{n}{2}$是一个非负整数，定义\[ f(n,p) = \min\{f(G) : \vert V(G)\vert = n, \ \vert E(G)\vert = p\binom{n}{2} \}.\] Erdés，Erdéczak和Spencer证明，对于$n \geq 2$，\[ (2n)^{\frac{1}{2}} - 2 \leq f(n, {\frac{1}{2}}) \leq 4n^{\frac{2}{3}}(\log n)^{\frac{1}{3}}.\]在本文中，我们证明了以下下界：对于$n^{-\frac{2}{3}} <p_n <1-n^{-\frac{1}{7}}$，\[ f(n,p) \geq \frac{1}{4}(1-p)^{\frac{2}{3}}n^{\frac{2}{3}} -1.\]此外，我们表明，这是紧到一个乘法常数因子附近的元素$\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\dots\}$的无穷多个$p$。与此相反，我们发现，对于任何$n$顶点图$G$，无论是$G$或$G^c$包含一个完整的子图上的$Ω(\frac{n}{\log n})$顶点。最后，我们讨论了随机图和伪随机图的满子图，以及几个公开问题。

英文摘要：

Let $G=(V,E)$ be a graph of density $p$ on $n$ vertices. Following Erdős, Łuczak and Spencer, an $m$-vertex subgraph $H$ of $G$ is called {\em full} if $H$ has minimum degree at least $p(m - 1)$. Let $f(G)$ denote the order of a largest full subgraph of $G$. If $p\binom{n}{2}$ is a non-negative integer, define \[ f(n,p) = \min\{f(G) : \vert V(G)\vert = n, \ \vert E(G)\vert = p\binom{n}{2} \}.\] Erdős, Łuczak and Spencer proved that for $n \geq 2$, \[ (2n)^{\frac{1}{2}} - 2 \leq f(n, {\frac{1}{2}}) \leq 4n^{\frac{2}{3}}(\log n)^{\frac{1}{3}}.\] In this paper, we prove the following lower bound: for $n^{-\frac{2}{3}} <p_n <1-n^{-\frac{1}{7}}$, \[ f(n,p) \geq \frac{1}{4}(1-p)^{\frac{2}{3}}n^{\frac{2}{3}} -1.\] Furthermore we show that this is tight up to a multiplicative constant factor for infinitely many $p$ near the elements of $\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\dots\}$. In contrast, we show that for any $n$-vertex graph $G$, either $G$ or $G^c$ contains a full subgraph on $Ω(\frac{n}{\log n})$ vertices. Finally, we discuss full subgraphs of random and pseudo-random graphs, and several open problems.

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：

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