A characterisation of edge-affine 2-arc-transitive covers of K_2ⁿ,2ⁿ

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’. To date, the vast majority of work in the literature has focused on classifying these ‘basic’ graphs. By contrast we give here a characterisation of the normal covers of the ‘basic’ 2-arc-transitive graphs K_2ⁿ,2ⁿ for n≥2. The characterisation identified the special role of graphs associated with a subgroup of automorphisms called an n-dimensional mixed dihedral group. This is a group H with two subgroups X and Y, each elementary abelian of order 2ⁿ, such that X∩Y=1, H is generated by X∪Y, and H/H^′≅X×Y. Our characterisation shows that each 2-arc-transitive normal cover of K_2ⁿ,2ⁿ 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, we show that the 2-arc-transitive group acting on the normal cover of K_2ⁿ,2ⁿ induces an edge-affine action on K_2ⁿ,2ⁿ (and we show that such actions are one of just four possible types of 2-arc-transitive actions on K_2ⁿ,2ⁿ). As a partial converse, we provide a graph theoretic characterisation of n-dimensional mixed dihedral groups, and finally, for each n≥2, we give an explicit construction of an n-dimensional mixed dihedral group which is a 2-group of order 2ⁿ²⁺²ⁿ, and a corresponding 2-arc-transitive normal cover of 2-power order of K_2ⁿ,2ⁿ. Note that these results partially address a problem proposed by Caiheng Li concerning normal covers of prime power order of the ‘basic’ 2-arc-transitive graphs.