Covers and pseudocovers of symmetric graphs

TL;DR

This paper introduces the concept of pseudocovers for symmetric graphs, providing a criterion for their identification and producing numerous new examples, expanding the understanding of symmetric graph extensions.

Contribution

It presents a new criterion for identifying pseudocovers of symmetric graphs and constructs multiple new examples, broadening the scope beyond previously known cases.

Findings

Most Praeger-Xu graphs are pseudocovers of wreath graphs.

Connected tetravalent symmetric graphs with certain stabilizer sizes have connected pseudocovers.

The paper extends the class of known pseudocovers significantly.

Abstract

We introduce the concept of pseudocover, which is a counterpart of cover, for symmetric graphs. The only known example of pseudocovers of symmetric graphs so far was given by Praeger, Zhou and the first-named author a decade ago, which seems technical and hard to extend to obtain more examples. In this paper, we present a criterion for a symmetric extender of a symmetric graph to be a pseudocover, and then apply it to produce various examples of pseudocovers, including (1) with a single exception, each Praeger-Xu's graph is a pseudocover of a wreath graph; (2) each connected tetravalent symmetric graph with vertex stabilizer of size divisible by $32$ has connected pseudocovers.