A characterisation of edge-affine 2-arc-transitive covers of \(\mathcal{K}_{2^n, 2^n}\) (Q6566000)

The family of finite \(2\)-arc-transitive graphs of a given valency is closed under forming non-trivial normal quotients, and graphs in this family having no non-trivial normal quotient are called basic. The authors in this paper give a characterisation of the normal covers of the basic \(2\)-arc-transitive graphs \(K_{2^n,2^n}\) for \(n \geq 2\). They show that each \(2\)-arc-transitive normal cover of \(K_{2^n,2^n}\) is either itself a Cayley graph, or is the line graph of a Cayley graph of an \(n\)-dimensional mixed dihedral group. In the latter case, the \(2\)-arc-transitive group acting on the normal cover of \(K_{2^n,2^n}\) is showed to induce an edge-affine action on \(K_{2^n,2^n}\). These results partially address a problem proposed by \textit{C. H. Li} [Bull. Lond. Math. Soc. 33, No. 2, 129--137 (2001; Zbl 1023.05074)] concerning normal covers of prime power order of the basic \(2\)-arc-transitive graphs.