当前位置: X-MOL 学术J. Graph Theory › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Canonical double covers of generalized Petersen graphs, and double generalized Petersen graphs
Journal of Graph Theory ( IF 0.9 ) Pub Date : 2020-11-03 , DOI: 10.1002/jgt.22642
Yan‐Li Qin 1 , Binzhou Xia 2 , Sanming Zhou 2
Affiliation  

The canonical double cover $\D(\Gamma)$ of a graph $\Gamma$ is the direct product of $\Gamma$ and $K_2$. If $\Aut(\D(\Gamma))\cong\Aut(\Gamma)\times\ZZ_2$ then $\Gamma$ is called stable; otherwise $\Gamma$ is called unstable. An unstable graph is said to be nontrivially unstable if it is connected, non-bipartite and no two vertices have the same neighborhood. In 2008 Wilson conjectured that, if the generalized Petersen graph $\GP(n,k)$ is nontrivially unstable, then both $n$ and $k$ are even, and either $n/2$ is odd and $k^2\equiv\pm 1 \pmod{n/2}$, or $n=4k$. In this note we prove that this conjecture is true. At the same time we determine all possible isomorphisms among the generalized Petersen graphs, the canonical double covers of the generalized Petersen graphs, and the double generalized Petersen graphs. Based on these we completely determine the full automorphism group of the canonical double cover of $\GP(n,k)$ for any pair of integers $n, k$ with $1 \leqslant k < n/2$.

中文翻译:

广义彼得森图的规范双覆盖和双广义彼得森图

图 $\Gamma$ 的规范双覆盖 $\D(\Gamma)$ 是 $\Gamma$ 和 $K_2$ 的直接乘积。如果 $\Aut(\D(\Gamma))\cong\Aut(\Gamma)\times\ZZ_2$ 那么 $\Gamma$ 被称为稳定的;否则 $\Gamma$ 称为不稳定。如果一个不稳定的图是连通的、非二部的并且没有两个顶点具有相同的邻域,则称其为非​​平凡不稳定的。在 2008 年,Wilson 推测,如果广义彼得森图 $\GP(n,k)$ 是非平凡不稳定的,那么 $n$ 和 $k$ 都是偶数,并且 $n/2$ 是奇数并且 $k^2 \equiv\pm 1 \pmod{n/2}$,或 $n=4k$。在这篇笔记中,我们证明这个猜想是正确的。同时,我们确定了广义彼得森图、广义彼得森图的规范双覆盖和双广义彼得森图之间所有可能的同构。
更新日期:2020-11-03
down
wechat
bug