Symmetric Covers and Pseudocovers of Complete Graphs

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 $\mathrm{\Gamma }$ is a pseudocover of its quotient $\mathrm{\Sigma }$ if they have the same valency and $\mathrm{\Gamma }$ is not a cover of $\mathrm{\Sigma }$.As the second result of this paper, we prove that the complete graph ${\text{K}}_n$ has a connected arc-transitive pseudocover if and only if $n-1$ is not a prime.