Sequentially embeddable graphs

Abstract

We call a (not necessarily planar) embedding of a graph $G$ in the plane \emph{sequential} if its vertices lie in ${\mathbb{Z}}^2$ and the line segments between adjacent vertices contain no interior integer points. In this note, we prove (i) a graph $G$ has a sequential embedding if and only if $G$ is 4-colorable, and (ii) if $G$ is planar, then $G$ has a sequential planar embedding.