Connectivity through bounds for the Castelnuovo-Mumford regularity

In this note we generalize and unify two results on connectivity of graphs: one by Balinsky and Barnette, one by Athanasiadis. This is done through a simple proof using commutative algebra tools. In particular we use bounds for the Castelnuovo-Mumford regularity of their Stanley-Reisner rings. As a result, if Δ is a simplicial d-pseudomanifold and s is the largest integer such that Δ has a missing face of size s, then the 1-skeleton of Δ is ⌈(s+1)ds⌉-connected. We also show that this value is tight.