Symmetrical 2-extensions of the 3-dimensional grid. I

Abstract

For a positive integer $d$, a connected graph $Γ$ is a symmetrical 2-extension of the $d$-dimensional grid $Λ^d$ if there exists a vertex-tran\-sitive group $G$ of automorphisms of $Γ$ and its imprimitivity system $σ$ with blocks of order 2 such that there exists an isomorphism $φ$ of the quotient graph $Γ/σ$ onto $Λ^d$. The tuple $(Γ, G, σ, φ)$ with specified components is called a realization of the symmetrical 2-extension $Γ$ of the grid $Λ^{d}$. Two realizations $(Γ_1, G_1,$ $σ_1, φ_1)$ and $(Γ_2, G_2, σ_2, φ_2)$ are called equivalent if there exists an isomorphism of the graph $Γ_1$ onto $Γ_2$ which maps $σ_1$ onto $σ_2$. V. Trofimov proved that, up to equivalence, there are only finitely many realizations of symmetrical $2$-extensions of $Λ^{d}$ for each positive integer $d$. E. Konovalchik and K. Kostousov found all, up to equivalence, realizations of symmetrical 2-extensions of the grid $Λ^2$. In this work we found all, up to equivalence, realizations $(Γ, G, σ, φ)$ of symmetrical 2-extensions of the grid $Λ^3$ for which only the trivial automorphism of $Γ$ preserves all blocks of $σ$ (we prove that there are 5573 such realizations, and that among corresponding graphs $Γ$ there are 5350 pairwise non-isomorphic).