A graph $X$ is said to be unstable if the direct product $X \times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is nontrivially unstable if it is unstable, connected, and nonbipartite, and no two distinct vertices of $X$ have exactly the same neighbors. We find three new conditions that each imply a circulant graph is unstable. (These yield infinite families of nontrivially unstable circulant graphs that were not previously known.) We also find all of the nontrivially unstable circulant graphs of order $2p$, where $p$ is any prime number. Our results imply that there does not exist a nontrivially unstable circulant graph of order $n$ if and only if either $n$ is odd, or $n < 8$, or $n = 2p$, for some prime number $p$ that is congruent to $3$ modulo $4$.
@article{10_37236_10655,
author = {Ademir Hujdurovi\'c and {\DJ}or{\dj}e Mitrovi\'c and Dave Witte Morris},
title = {On automorphisms of the double cover of a circulant graph},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {4},
doi = {10.37236/10655},
zbl = {1486.05129},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10655/}
}
TY - JOUR
AU - Ademir Hujdurović
AU - Đorđe Mitrović
AU - Dave Witte Morris
TI - On automorphisms of the double cover of a circulant graph
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/10655/
DO - 10.37236/10655
ID - 10_37236_10655
ER -
%0 Journal Article
%A Ademir Hujdurović
%A Đorđe Mitrović
%A Dave Witte Morris
%T On automorphisms of the double cover of a circulant graph
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/10655/
%R 10.37236/10655
%F 10_37236_10655
Ademir Hujdurović; Đorđe Mitrović; Dave Witte Morris. On automorphisms of the double cover of a circulant graph. The electronic journal of combinatorics, Tome 28 (2021) no. 4. doi: 10.37236/10655
