Abstract
Abstract Tverberg’s theorem states that a set with sufficiently many points in $${\mathbb {R}}^d$$ R d can always be partitioned into m parts such that the nerve (the intersection pattern) of the convex hulls of the parts form an $$(m-1)$$ ( m - 1 ) -simplex. De Loera, Hogan, Oliveros, and Yang (2021) explored how other simplicial complexes can emerge as nerve complexes for sufficiently large point sets. In this paper, we establish a connection between the theory of word-representable graphs and a method for encoding the 1-skeletons of simplicial complexes to generate nerve complexes. Specifically, we demonstrate that every triangle-free 2-word-representable graph can be realized as a nerve complex in the plane, given sufficiently many points. Furthermore, for every bipartite graph, there exists a dimension d such that it can be represented as a nerve complex for sufficiently many points in $${\mathbb {R}}^d$$ R d .
Published Version
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