Box representations of embedded graphs

TL;DR

This paper extends the concept of box representations of graphs to those embedded in surfaces, establishing bounds on the box dimension based on genus and cycle length, with implications for graph theory and geometric representations.

Contribution

It proves that graphs on fixed surfaces without short non-contractible cycles have 5-box representations and introduces a linear function relating genus to vertex removal for such representations.

Findings

Graphs on fixed surfaces without short cycles have 5-box representations.

A linear function bounds the number of vertices to remove for genus-based graphs.

Graphs in minor-closed classes without short cycles have 3-box representations.

Abstract

A $d$-box is the cartesian product of $d$ intervals of $\mathbb{R}$ and a $d$-box representation of a graph $G$ is a representation of $G$ as the intersection graph of a set of $d$-boxes in $\mathbb{R}^d$. It was proved by Thomassen in 1986 that every planar graph has a 3-box representation. In this paper we prove that every graph embedded in a fixed orientable surface, without short non-contractible cycles, has a 5-box representation. This directly implies that there is a function $f$, such that in every graph of genus $g$, a set of at most $f(g)$ vertices can be removed so that the resulting graph has a 5-box representation. We show that such a function $f$ can be made linear in $g$. Finally, we prove that for any proper minor-closed class $\mathcal{F}$, there is a constant $c(\mathcal{F})$ such that every graph of $\mathcal{F}$ without cycles of length less than $c(\mathcal{F})$ has a 3-box representation, which is best possible.