Boxicity of graphs on surfaces

TL;DR

This paper establishes upper bounds on the boxicity of graphs embedded on surfaces, showing that toroidal graphs have boxicity at most 7 and graphs on surfaces of genus g have boxicity at most 5g+3, improving understanding of their geometric representations.

Contribution

The paper provides new upper bounds on the boxicity of graphs on surfaces, extending previous results for planar and outerplanar graphs to more complex surfaces.

Findings

Boxicity of toroidal graphs is at most 7.

Boxicity of graphs on a surface of genus g is at most 5g+3.

Improved bounds on the dimension of the adjacency poset of surface-embedded graphs.

Abstract

The boxicity of a graph $G=(V,E)$ is the least integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap ... \cap E_k$. Scheinerman proved in 1984 that outerplanar graphs have boxicity at most two and Thomassen proved in 1986 that planar graphs have boxicity at most three. In this note we prove that the boxicity of toroidal graphs is at most 7, and that the boxicity of graphs embeddable in a surface $\Sigma$ of genus $g$ is at most $5g+3$. This result yields improved bounds on the dimension of the adjacency poset of graphs on surfaces.