Better bounds for poset dimension and boxicity

TL;DR

This paper establishes near-optimal bounds on the dimension of posets and boxicity of graphs with maximum degree, improving longstanding results and solving an open problem related to graphs with Euler genus.

Contribution

It provides improved upper bounds on poset dimension and graph boxicity based on maximum degree, and determines the maximum boxicity for graphs with given Euler genus.

Findings

Poset dimension is at most Δ log^{1+o(1)} Δ for graphs with maximum degree Δ.

Graph boxicity is at most Δ log^{1+o(1)} Δ for graphs with maximum degree Δ.

Maximum boxicity of graphs with Euler genus g is Θ(√(g log g)), solving an open problem.

Abstract

We prove that the dimension of every poset whose comparability graph has maximum degree $\Delta$ is at most $\Delta\log^{1+o(1)} \Delta$. This result improves on a 30-year old bound of F\"uredi and Kahn, and is within a $\log^{o(1)}\Delta$ factor of optimal. We prove this result via the notion of boxicity. The "boxicity" of a graph $G$ is the minimum integer $d$ such that $G$ is the intersection graph of $d$-dimensional axis-aligned boxes. We prove that every graph with maximum degree $\Delta$ has boxicity at most $\Delta\log^{1+o(1)} \Delta$, which is also within a $\log^{o(1)}\Delta$ factor of optimal. We also show that the maximum boxicity of graphs with Euler genus $g$ is $\Theta(\sqrt{g \log g})$, which solves an open problem of Esperet and Joret and is tight up to a $O(1)$ factor.