Representability and boxicity of simplicial complexes

TL;DR

This paper introduces the concept of $d$-boxicity for simplicial complexes, generalizing graph boxicity, and establishes bounds on this measure based on the structure of missing faces, with exact values for specific systems.

Contribution

It defines $d$-boxicity for simplicial complexes and proves bounds relating it to missing face structures, extending the concept of boxicity from graphs to higher-dimensional complexes.

Findings

Bound on $d$-boxicity for complexes without large missing faces

Exact $d$-boxicity for complexes with Steiner systems as missing faces

Generalization of graph boxicity to higher dimensions

Abstract

Let $X$ be a simplicial complex on vertex set $V$. We say that $X$ is $d$-representable if it is isomorphic to the nerve of a family of convex sets in $\mathbb{R}^d$. We define the $d$-boxicity of $X$ as the minimal $k$ such that $X$ can be written as the intersection of $k$ $d$-representable simplicial complexes. This generalizes the notion of boxicity of a graph, defined by Roberts. A missing face of $X$ is a set $\tau\subset V$ such that $\tau\notin X$ but $\sigma\in X$ for any $\sigma\subsetneq \tau$. We prove that the $d$-boxicity of a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$ is at most $\left\lfloor\frac{1}{d+1}\binom{n}{d}\right\rfloor$. The bound is sharp: the $d$-boxicity of a simplicial complex whose set of missing faces form a Steiner $(d,d+1,n)$-system is exactly $\frac{1}{d+1}\binom{n}{d}$.