From Word-Representable Graphs to Altered Tverberg-Type Theorems
Por:
Oliveros, D, Torres, AJ
Publicada:
1 mar 2025
Ahead of Print:
1 feb 2025
Resumen:
Tverberg's theorem states that a set with sufficiently many points in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}<^>d$$\end{document} 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m-1)$$\end{document}-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 Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}<^>d$$\end{document}.
Filiaciones:
Oliveros, D:
Univ Nacl Autonoma Mexico, Inst Matemat, Juriquilla 3001, Santiago De Queretaro 76230, Queretaro, Mexico
Torres, AJ:
Univ Calif Davis, Dept Math, 1 Shields Ave, Davis, CA 95616 USA
Green Submitted, hybrid, All Open Access; Green Open Access; Hybrid Gold Open Access
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