Abstract
We investigate the question of whether any d-colorable simplicial d-polytope can be octahedralized, i.e., can be subdivided to a d-dimensional geometric cross-polytopal complex. We give a positive answer in dimension 3, with the additional property that the octahedralization introduces no new vertices on the boundary of the polytope.
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Acknowledgements
The first author was supported by the Einstein Foundation Berlin. The second author was supported by the German Research Council DFG GRK-1916. We would like to thank Martina Juhnke-Kubitzke for suggesting the problem to us and for insightful discussions and comments on the manuscript. Many thanks to Francisco Santos for interesting discussions and comments on the manuscript, and in particular pointing out the decomposition of the octahedron as the Schlegel diagram of the 24-cell. We thank Eran Nevo for carefully listening to our presentation and suggesting the simplification of Remark 4.7, which allows us to give Theorem 1.7 in its final form. A thank you also to Hannah Sjöberg for pointing out Lemma 4.2. A further thank you to the anonymous reviewers for the many helpful suggestions which made the paper easier to read.
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Codenotti, G., Venturello, L. Octahedralizing 3-Colorable 3-Polytopes. Discrete Comput Geom 66, 1429–1445 (2021). https://doi.org/10.1007/s00454-020-00262-4
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DOI: https://doi.org/10.1007/s00454-020-00262-4