On the connectivity of manifold graphs

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.