乘积图的诱导子图的最小可能最大度
On the Minimum Possible Maximum Degree of Induced Subgraphs of Product Graphs
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中文总结 AI 辅助
本文研究乘积图的诱导子图的最大度问题，确定了Hamming乘积和张量乘积中指定大小诱导子图的最小可能平均度，并证明张量乘积具有Huang类现象，且结果可推广至完全图。
中文摘要 AI 辅助
以下是一个关于图$G$的自然且基本的问题：如果$G$的一个诱导子图$H$比最大独立集多$x$个顶点，那么$H$的最大度作为$x$的函数可以如何描述？$x=1$的情况已经引起了相当大的兴趣。例如，在他著名的敏感性猜想证明中，Hao Huang证明了超立方体$Q_n$的任意具有超过$2^{n-1}$个顶点的诱导子图的最大度至少为$\sqrt{n}$。Chung、Fúredi、Graham和Seymour证明了该界是紧的。在本文中，我们研究当$G$是三角形的$n$重Hamming乘积或$n$重张量乘积时的问题。对于这两个图，我们确定了指定大小的诱导子图的平均度的精确最小可能值。我们还证明了张量乘积表现出若干Huang类现象。对于Hamming乘积，我们证明，对于若干大小密度和大$n$，最小可能最大度渐近等于最小可能平均度。最后，我们将若干结果从三角形推广到任意完全图$K_k$。
英文摘要
The following is a natural and fundamental question for a graph $G$: if an induced subgraph $H$ of $G$ has $x$ more vertices than a maximum independent set, what can be said about the maximum degree of $H$ as a function of $x$? The case $x=1$ is already of considerable interest. For example, in his celebrated proof of the sensitivity conjecture, Hao Huang showed that every induced subgraph of the hypercube $Q_n$ on more than $2^{n-1}$ vertices has maximum degree at least $\sqrt{n}$. Chung, Fúredi, Graham, and Seymour proved that this bound is tight. In this paper, we study this question when $G$ is either the $n$-fold Hamming product or the $n$-fold tensor product of a triangle. For both graphs, we determine the exact minimum possible average degree of an induced subgraph of a prescribed size. We also prove that the tensor product exhibits several Huang-like phenomena. For the Hamming product, we show that, for several size densities and large $n$, the minimum possible maximum degree is asymptotically equal to the minimum possible average degree. Finally, we extend several of the results from the triangle to an arbitrary complete graph $K_k$.
发表机构
- Weizmann Institute of Science（魏茨曼科学研究所）
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