A family of multiply extended grids (Q1923782)

The paper deals with an infinite family \((\Gamma_n)^\infty_{n=5}\) of finite connected graphs \(\Gamma_n\) that are multiple extensions of the well-known ``extended grid,'' a locally 4-by-4 grid graph discovered by \textit{A. Blokhuis} and \textit{A. E. Brouwer} [J. Graph Theory 13, No. 2, 229-244 (1989; Zbl 0722.05054)]. For \(n>5\), the graphs \(\Gamma_n\) are locally \(\Gamma_{n-1}\) and have the following properties: the automorphism group \(G(n)\) of \(\Gamma_n\) acts transitively on the maximal cliques (which are of order \(n\)), and the stabilizer of some \(n\)-clique \(\pi\) of \(\Gamma_n\) induces \(\Sigma_n\) on the vertices of \(\pi\). The clique complex of \(\Gamma_n\) is shown to be simply connected.