From word-representable graphs to altered Tverberg-type theorems

TL;DR

This paper explores how various classes of graphs can be realized as nerve complexes of point sets in Euclidean space, linking word-representable graphs to Tverberg-type theorems and convex geometry.

Contribution

It establishes that multiple graph classes, including triangle-free, circle, outerplanar, and bipartite graphs, can be induced as nerve complexes in Euclidean space, connecting graph theory and convex geometry.

Findings

Triangle-free, circle, outerplanar, and bipartite graphs can be realized as nerve complexes.

Word-representable graphs encode 1-skeletons of simplicial complexes.

Sets of points in Euclidean space can generate these nerve complexes.

Abstract

Tverberg's theorem says that a set with sufficiently many points in $\mathbb{R}^d$ can always be partitioned into $m$ parts so that the $(m-1)$-simplex is the (nerve) intersection pattern of the convex hulls of the parts. In arXiv:1808.00551v1 [math.MG] the authors investigate how other simplicial complexes arise as nerve complexes once we have a set with sufficiently many points. In this paper we relate the theory of word-representable graphs as a way of codifying $1$-skeletons of simplicial complexes to generate nerves. In particular, we show that every $2$-word-representable triangle-free graph, every circle graph, every outerplanar graph, and every bipartite graph could be induced as a nerve complex once we have a set with sufficiently many points in $\mathbb{R}^d$ for some $d$.