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 bM for simplicial d-manifolds M taking values in the range 0 <= bM <= d-1. The main result is that bM influences connectivity in the following way: The graph of a d-dimensional simplicial compact manifold M is (2d - bM)-connected. The parameter bM has the property that bM = 0 if the complex M 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 bM involves the concept of banner triangulations of manifolds, a generalization of flag triangulations.
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