$$l_{1}$$ -embeddability of shifted quadrilateral cylinder graphs

Abstract

A connected graph G is called \(l_{1}\)-embeddable, if it can be isometrically embedded into the \(l_{1}\)-space. The shifted quadrilateral cylinder graph \(O_{m,n,k}\) is a class of quadrilateral tilings on a cylinder obtained by rolling the grid graph \(P_{m}\square P_{n}\) via shifting k positions. In this article, we determine that all the \(O_{m,n,k}\) are not \(l_{1}\)-embeddable except for \(O_{m,n,0}\) and \(O_{m,3,1}\).