从另一个角度看更奇的图论
More odd graph theory from another point of view
摘要：
Kneser图$K(n, k)$具有$[n]=\{1,2,...,n \}$的所有$k$ -元素子集作为顶点，以及不相交的任何两个顶点之间的边。如果$n=2k+1$，则$K(n, k)$称为奇图。设$ n >4$和$1< k < \frac{n}{2} $。本文证明了：如果Kneser图$K(n,k)$是偶数阶的，其中$n$是奇整数或两个整数$n,k $都是偶数，则$K(n,k)$是点传递非Cayley图.虽然，这些是Godsil [8]的特殊情况，不像他的证明使用了一些非常深刻的群论事实，我们的证明没有使用沉重的群论事实。我们利用数论和群论中的一些相当基本的事实得到了这些结果。我们证明了“几乎所有”奇图都是偶阶的，因此是点传递的非Cayley图。最后，证明了如果$ k > 4 $是一个偶数，使得对于某些$ t> 2 $，$k$不具有$ k = 2^t $的形式，则奇图$O_{k+1}$的线图是一个点传递的非Cayley图.
英文摘要：
The Kneser graph $K(n, k)$ has as vertices all $k$-element subsets of $[n]=\{1,2,...,n \}$ and an edge between any two vertices that are disjoint. If $n=2k+1$, then $K(n, k)$ is called an odd graph. Let $ n >4$ and $1< k < \frac{n}{2} $. In the present paper, we show that if the Kneser graph $K(n,k)$ is of even order where $n$ is an odd integer or both of the integers $n,k $ are even, then $K(n,k)$ is a vertex-transitive non Cayley graph. Although, these are special cases of Godsil [8], unlike his proof that uses some very deep group-theoretical facts, ours uses no heavy group-theoretic facts. We obtain our results by using some rather elementary facts of number theory and group theory. We show that "almost all" odd graphs are of even order, and consequently are vertex-transitive non Cayley graphs. Finally, we show that if $ k > 4 $ is an even integer such that $k$ is not of the form $ k = 2^t $ for some $ t> 2 $, then the line graph of the odd graph $O_{k+1}$ is a vertex-transitive non Cayley graph.
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。
