From word-representable graphs to altered Tverberg-type theorems

11/19/2021

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by   Deborah Oliveros, et al.

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Tverberg's theorem says that a set with sufficiently many points in 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 ℝ^d for some d.