# Sequentially embeddable graphs

## TL;DR

This paper characterizes graphs with sequential embeddings in the plane, linking their existence to 4-colorability, and shows all planar graphs admit such embeddings.

## Contribution

It establishes a precise equivalence between sequential embeddings and 4-colorability, and proves all planar graphs can be embedded sequentially in the plane.

## Key findings

- A graph has a sequential embedding iff it is 4-colorable.
- All planar graphs have a sequential planar embedding.

## 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.