Many Turán exponents via subdivisions
Many Turán exponents via subdivisions
复制标题
许多图兰指数通过细分
DOI：
--
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Y. Qiu
中科院分区：
文献类型：
--
作者：
T. Jiang;Y. Qiu
<jats:p>给定一个图<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline1.png" />
<jats:tex-math>
$H$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>和一个正整数<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline2.png" />
<jats:tex-math>
$n$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula><jats:italic>图兰数</jats:italic><jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline3.png" />
<jats:tex-math>
$mathrm{ex}（n，H）$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>中的最大边数<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline4.png" />
<jats:tex-math>
$n$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>- 顶点图，不包含<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline5.png" />
<jats:tex-math>
$H$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>作为子图。一个真实的数字<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline6.png" />
<jats:tex-math>
$rin（1，2）$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>称为<jats:italic>Turán指数</jats:italic>，如果存在一个二部图<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline7.png" />
<jats:tex-math>
$H$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>使得<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline8.png" />
<jats:tex-math>
$mathrm{ex}（n，H）=Theta（n^r）$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>.一个长期存在的猜想Erdos和Simonovits国家，<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline9.png" />
<jats:tex-math>
$1+压裂{p}{q}$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>对所有正整数都是图兰指数<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline10.png" />
<jats:tex-math>
$p$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>和<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline11.png" />
<jats:tex-math>
$q$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>与<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline12.png" />
<jats:tex-math>
$qgt p$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. </jats:p>
<jats:p>在本文中，我们表明，<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline13.png" />
<jats:tex-math>
$1+压裂{p}{q}$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>对所有正整数都是图兰指数<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline14.png" />
<jats:tex-math>
$p$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>和<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline15.png" />
<jats:tex-math>
$q$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>与<jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline16.png" />
<jats:tex-math>
$q gt p^{2}$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>.我们的结果也解决了Janzer [18]的一个猜想。</jats:p>
<jats:p>Given a graph <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline1.png" />
<jats:tex-math>
$H$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> and a positive integer <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline2.png" />
<jats:tex-math>
$n$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, the <jats:italic>Turán number</jats:italic><jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline3.png" />
<jats:tex-math>
$mathrm{ex}(n,H)$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is the maximum number of edges in an <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline4.png" />
<jats:tex-math>
$n$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>-vertex graph that does not contain <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline5.png" />
<jats:tex-math>
$H$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> as a subgraph. A real number <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline6.png" />
<jats:tex-math>
$rin (1,2)$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is called a <jats:italic>Turán exponent</jats:italic> if there exists a bipartite graph <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline7.png" />
<jats:tex-math>
$H$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> such that <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline8.png" />
<jats:tex-math>
$mathrm{ex}(n,H)=Theta (n^r)$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. A long-standing conjecture of Erdős and Simonovits states that <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline9.png" />
<jats:tex-math>
$1+frac{p}{q}$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is a Turán exponent for all positive integers <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline10.png" />
<jats:tex-math>
$p$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> and <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline11.png" />
<jats:tex-math>
$q$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> with <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline12.png" />
<jats:tex-math>
$qgt p$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>.</jats:p>
<jats:p>In this paper, we show that <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline13.png" />
<jats:tex-math>
$1+frac{p}{q}$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is a Turán exponent for all positive integers <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline14.png" />
<jats:tex-math>
$p$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> and <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline15.png" />
<jats:tex-math>
$q$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> with <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000177_inline16.png" />
<jats:tex-math>
$q gt p^{2}$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. Our result also addresses a conjecture of Janzer [18].</jats:p>
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影响因子：
0.8
作者：
Janzer O
通讯作者：
Janzer O
关于有理图兰指数猜想
DOI：
10.1016/j.jctb.2020.12.003
发表时间：
2021
期刊：
Journal of Combinatorial Theory, Series B
影响因子：
--
作者：
Kang D
通讯作者：
Kang D
影响因子：
0.9
作者：
Janzer O
通讯作者：
Janzer O
影响因子：
0.8
作者：
Jiang, Tao;Qiu, Yu
通讯作者：
Qiu, Yu