Symmetric Covers and Pseudocovers of Complete Graphs

TL;DR

This paper characterizes all faithful arc-transitive covers of complete graphs, introduces the concept of pseudocovers, and determines when complete graphs have connected arc-transitive pseudocovers based on primality conditions.

Contribution

It provides a complete characterization of faithful arc-transitive covers and introduces the concept of pseudocovers, establishing conditions for their existence in complete graphs.

Findings

Complete graphs have connected arc-transitive pseudocovers if and only if n-1 is not prime.

A general construction method for faithful arc-transitive covers of complete graphs.

Characterization of all faithful arc-transitive covers of complete graphs.

Abstract

We first characterize all faithful arc-transitive covers of complete graphs and we give a general construction of such covers. For a counterpart of cover, we say a graph $\Gamma$ is a pseudocover of its quotient $\Sigma$ if they have the same valency and $\Gamma$ is not a cover of $\Sigma$.As the second result of this paper, we prove that the complete graph $\K_n$ has a connected arc-transitive pseudocover if and only if $n-1$ is not a prime.