$l_1$—Embeddability of Hexagonal and Quadrilateral Mobius graphs

Abstract

A connected graph $G$ is called $l_1$-embeddable, if $G$ can be isometrically embedded into the $i$-space. The hexagonal Möbius graphs $H_{2m,2k}$ and $H_{2m+1,2k+1}$ are two classes of hexagonal tilings of a Möbius strip. The regular quadrilateral Möbius graph $Q_{p,q}$ is a quadrilateral tiling of a Möbius strip. In this note, we show that among these three classes of graphs only $H_{2,2}$, $H_{3,3}$, and $Q_{2,2}$ are $l_1$-embeddable.