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On the connectivity of manifold graphs


Authors: Anders Björner and Kathrin Vorwerk
Journal: Proc. Amer. Math. Soc. 143 (2015), 4123-4132
MSC (2010): Primary 05E45; Secondary 05C40
DOI: https://doi.org/10.1090/proc/12415
Published electronically: June 18, 2015
MathSciNet review: 3373913
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Abstract: This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant $ b_{\Delta }$ of a simplicial $ d$-manifold $ \Delta $ taking values in the range $ 0\le b_{\Delta } \le d-1$. The main result is that $ b_\Delta $ influences connectivity in the following way: The graph of a $ d$-dimensional simplicial compact manifold $ \Delta $ is $ (2d-b_{\Delta })$-connected.

The parameter $ b_{\Delta }$ has the property that $ b_{\Delta } =0$ if the complex $ \Delta $ is flag. Hence, our result interpolates between Barnette's theorem (1982) that all $ d$-manifold graphs are $ (d+1)$-connected and Athanasiadis' theorem (2011) that flag $ d$-manifold graphs are $ 2d$-connected.

The definition of $ b_{\Delta }$ involves the concept of banner triangulations of manifolds, a generalization of flag triangulations.


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Additional Information

Anders Björner
Affiliation: Royal Institute of Technology, Department of Mathematics, S-100 44 Stockholm, Sweden
Email: bjorner@math.kth.se

Kathrin Vorwerk
Affiliation: Royal Institute of Technology, Department of Mathematics, S-100 44 Stockholm, Sweden
Email: vorwerk@math.kth.se

DOI: https://doi.org/10.1090/proc/12415
Received by editor(s): July 23, 2012
Received by editor(s) in revised form: August 20, 2012, October 22, 2013, and November 19, 2013
Published electronically: June 18, 2015
Additional Notes: This research was supported by the Knut and Alice Wallenberg Foundation, grant KAW.2005.0098
Communicated by: Jim Haglund
Article copyright: © Copyright 2015 American Mathematical Society

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