On the connectivity of manifold graphs
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- by Anders Björner and Kathrin Vorwerk PDF
- Proc. Amer. Math. Soc. 143 (2015), 4123-4132 Request permission
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.
References
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Additional Information
- Anders Björner
- Affiliation: Royal Institute of Technology, Department of Mathematics, S-100 44 Stockholm, Sweden
- MR Author ID: 37500
- 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
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 4123-4132
- MSC (2010): Primary 05E45; Secondary 05C40
- DOI: https://doi.org/10.1090/proc/12415
- MathSciNet review: 3373913