Polyhexes that are \(\ell _1\) graphs (Q1024308)

A graph \(\Gamma\) is an \(\ell_1\) graph if the vertices of \(\Gamma\) can be matched with some points of \(\ell_1\) in such a way that the path distance between every two vertices of \(\Gamma\) is equal to the distance between the corresponding two points of \(\ell_1\). A polyhex is a finite trivalent surface graph \(\Gamma\) drawn on a surface \(S\) in such a way that every face is a hexagon. The authors determine all polyhexes that are \(\ell_1\) as follows: If \(\Gamma\) is an \(\ell_1\) polyhex then either \(\Gamma\) is the so-called prism polyhex \(\Gamma_{2n}\), \(n\geq 3\), or one of the four sporadic examples.