Abstract
The topological Tverberg theorem states that for any prime power q and continuous map from a (d+1)(q−1)-simplex to ℝd, there are q disjoint faces F i of the simplex whose images intersect. It is possible to put conditions on which pairs of vertices of the simplex that are allowed to be in the same face F i . A graph with the same vertex set as the simplex, and with two vertices adjacent if they should not be in the same F i , is called a Tverberg graph if the topological Tverberg theorem still work.
These graphs have been studied by Hell, Schöneborn and Ziegler, and it is known that disjoint unions of small paths, cycles, and complete graphs are Tverberg graphs. We find many new examples by establishing a local criterion for a graph to be Tverberg. An easily stated corollary of our main theorem is that if the maximal degree of a graph is D, and D(D+1)<q, then it is a Tverberg graph.
We state the affine versions of our results and also describe how they can be used to enumerate Tverberg partitions.
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References
I. Bárány, S.B. Shlosman, A. Szűcs: On a topological generalization of a theorem of Tverberg, J. London Math. Soc. 23(2) (1981), 158–164.
P.V.M. Blagojević, B. Matschke, G.M. Ziegler: Optimal bounds for the colored Tverberg problem, arxiv:0910.4987.
P.V.M. Blagojević, B. Matschke, G.M. Ziegler: Optimal bounds for a colorful Tverberg-Vrecica type problem, Adv. Math. 226 (2011), 5198–5215.
A. Björner: Topological Methods, in: “Handbook of Combinatorics” (eds. R. Graham, M. Grötschel, and L. Lovász), North-Holland, 1995, 1819–1872.
A. Björner, L. Lovász, R. Živaljević, S. Vrećica: Chessboard complexes and matching complexes, J. London Math. Soc. 49 (1994), 25–39.
J.A. Bondy, U.S.R. Murty: Graph theory, Graduate Texts in Mathematics, 244. Springer, New York, 2008.
A. Dochtermann, A. Engström: Algebraic properties of edge ideals via combinatorial topology. Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of Anders Björner, Research Paper 2, 24.
A. Engström: Independence complexes of claw-free graphs, European J. Combin. 29 (2008), no. 1, 234–241.
A. Engström, P. Norén: Tverberg’s theorem and graph coloring, arxiv: 1105.1455.
S. Hell: On the number of Tverberg partitions in the prime power case, European J. Combin. 28 (2007), 347–355.
S. Hell: Tverberg’s theorem with constraints, J. Combin. Theory, Ser. A 115 (2008), 1402–1416.
J. Matoušek: Using the Borsuk-Ulam theorem, Springer-Verlag, Berlin, 2nd corrected priting, 2008.
K.S. Sarkaria: Tverberg partitions and Borsuk-Ulam theorems, Pacific J. Math. 196 (2000), 231–241.
T. Schöneborn, G.M. Ziegler: The Topological Tverberg Theorem and winding numbers, J. Combin. Theory, Ser. A 112 (2005), 82–104.
H. Tverberg: A generalization of Radon’s theorem, J. London Math. Soc. 41 (1966), 123–128.
A.Y. Volovikov: On a topological generalization of Tverberg’s theorem, Mat. Zametski 59 (1996), 454–456 (Translation: Math. Notes. 59 3–4 (1996), 324–325).
A. Vrećica, R. Živaljević: The colored Tverberg’s problem and complexes of injective functions, J. Combin. Theory, Ser. A 22 (1990), 183–186.
A. Vučić, R. Živaljević: Note on a conjecture of Sierksma, Discrete Comput. Geom. 9 (1993), no. 4, 339–349.
G.M. Ziegler: Shellability of chessboard complexes. Israel J. Math. 87 (1994), no. 1–3, 97–110.
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The author is a Miller Research Fellow 2009–2012 at UC Berkeley, and gratefully acknowledges support from the Adolph C. and Mary Sprague Miller Institute for Basic Research in Science.
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Engström, A. A local criterion for Tverberg graphs. Combinatorica 31, 321–332 (2011). https://doi.org/10.1007/s00493-011-2665-9
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DOI: https://doi.org/10.1007/s00493-011-2665-9