Metacirculants and split weak metacirculants
Abstract
Metacirculants are a rich resource of many families of interesting graphs, and weak metacirculants are generalizations of them. A graph is called a {\em split weak metacirculant} if it has a vertex-transitive split metacyclic automorphism group. In two recent papers, it is shown that a graph of prime power order is a metacirculant if and only if it is a split weak metacirculant. Let is a positive integer. In this paper, we first give a sufficient condition for the existence of split weak metacirculants of order which are not metacirculants. This is then used to give a sufficient and necessary condition for the existence of split weak metacirculants of order which are not metacirculants, where is a product of two prime-powers. As byproducts, we construct infinitely many split weak metacirculant graphs which are not metacirculant graphs, and answer an open question reported in the literature.
Cite
@article{arxiv.1804.01632,
title = {Metacirculants and split weak metacirculants},
author = {Li Cui and Jin-Xin Zhou},
journal= {arXiv preprint arXiv:1804.01632},
year = {2018}
}
Comments
We find that there are some flaws in the proof of the main theorem. More precisely, the flaw is in the proof of the necessary condition for the existence of split weak metacirculants of order $n$ which are not metacirculants, where $n$ is a product of two prime-powers. We will try to find some new method to correct these flaws