On the connectivity of manifold graphs

TL;DR

This paper establishes a lower bound on the connectivity of graphs derived from triangulated compact manifolds, introducing a new invariant that interpolates between known connectivity bounds based on flag properties.

Contribution

Introduces the invariant b_M for simplicial manifolds, linking it to graph connectivity and generalizing previous theorems for flag and non-flag manifolds.

Findings

Graph connectivity is at least 2d - b_M for a d-manifold.

b_M = 0 for flag manifolds, achieving maximum connectivity.

Connects manifold triangulation properties with graph connectivity bounds.

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