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arXiv 2203.14036math.CO
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关于广义Kneser图的树宽

On the treewidth of generalized Kneser graphs

Klaus Metsch

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摘要：

整数$k>t>0$和$n>2k-t$的广义Kneser图$K(n,k,t)$是其顶点是$\{1,\dots,n\}$的$k$ -子集且两个顶点相邻的图当且仅当它们共享少于$t$的元素。当$t\ge 2$和$n$与$k$相比足够大时，我们确定了广义Kneser图$K(n,k,t)$的树宽.对$n$施加的界限是对先前已知界限的显著改进。我们结果的一个推论如下。对于每个整数$c\ge 1$，存在一个常数$K(c)\ge 2c$，使得$k\ge K(c)$对于$t=k-c$意味着$$tw(K(n,k,t))=\binom{n}{k}-\binom{n-t}{k-t}-1$$当且仅当$n\ge (t+1)(k+1-t)$。

英文摘要：

The generalized Kneser graph $K(n,k,t)$ for integers $k>t>0$ and $n>2k-t$ is the graph whose vertices are the $k$-subsets of $\{1,\dots,n\}$ with two vertices adjacent if and only if they share less than $t$ elements. We determine the treewidth of the generalized Kneser graphs $K(n,k,t)$ when $t\ge 2$ and $n$ is sufficiently large compared to $k$. The imposed bound on $n$ is a significant improvement of a previously known bound. One consequence of our result is the following. For each integer $c\ge 1$ there exists a constant $K(c)\ge 2c$ such that $k\ge K(c)$ implies for $t=k-c$ that $$tw(K(n,k,t))=\binom{n}{k}-\binom{n-t}{k-t}-1$$ if and only if $n\ge (t+1)(k+1-t)$ .

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