具有超线性盈余和非并极值图的有限禁用族
A finite forbidden family with superlinear surplus and non-join extremal graphs
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中文总结 AI 辅助
该研究构造了满足超线性盈余的有限禁用族，给出极值图论中两个乘积结构猜想的共同反例，其极值图补图连通、无并分解，推翻了相关猜想。
中文摘要 AI 辅助
我们给出了极值图论中两个乘积结构猜想的共同反例。更确切地说，我们构造了一个固定的非空有限族$\boldsymbol{\textit{L}}$，满足$p(\boldsymbol{\textit{L}})=2$，使得对任意足够大的$n$，存在$c>0$使得$\boldsymbol{\text{ex}}(n,\boldsymbol{\textit{L}})>t_2(n)+cn^{3/2}$。尽管如此，在每个这样的阶数下都存在一个$\boldsymbol{\textit{L}}$极值图，其补图是连通的，因此没有非平凡的并分解。这种超线性盈余还迫使$\boldsymbol{\textit{L}}$的分解族不包含任何森林。该构造使用了一种端点单射修复操作，其有限障碍族的极值数及等式情形可精确描述，这些性质推翻了两个猜想。
英文摘要
We give a common counterexample to two product-structure conjectures in extremal graph theory. More precisely, we construct a fixed nonempty finite family $\mathcal L$ with $p(\mathcal L)=2$ such that, for some $c>0$, \[ \operatorname{ex}(n,\mathcal L)>t_2(n)+cn^{3/2} \] for every sufficiently large $n$. Nevertheless, at every such order there is an $\mathcal L$-extremal graph whose complement is connected and which therefore admits no decomposition as a join of two nonempty graphs. This superlinear surplus also forces the decomposition family of $\mathcal L$ to contain no forest. The construction uses endpoint-injective repair with a finite obstruction family admitting an exact extremal formula and equality classification.
发表机构
- School of Mathematics and Statistics, Xidian University（西安电子科技大学数学与统计学院）
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