On the Connectivity of Fiber Graphs

TL;DR

This paper investigates the connectivity properties of fiber graphs related to Gr"obner and Graver basis moves, providing counterexamples to existing conjectures and establishing optimal connectivity conditions.

Contribution

It constructs fiber graphs with minimal edge-connectivity using Gr"obner basis moves and proves that Graver basis moves ensure optimal connectivity, challenging previous assumptions.

Findings

Fiber graphs with Gr"obner basis moves can have minimal edge-connectivity.

Graph properties are independent of the right-hand side size.

Graver basis moves guarantee optimal edge-connectivity.

Abstract

We consider the connectivity of fiber graphs with respect to Gr\"obner basis and Graver basis moves. First, we present a sequence of fiber graphs using moves from a Gr\"obner basis and prove that their edge-connectivity is lowest possible and can have an arbitrarily large distance from the minimal degree. We then show that graph-theoretic properties of fiber graphs do not depend on the size of the right-hand side. This provides a counterexample to a conjecture of Engstr\"om on the node-connectivity of fiber graphs. Our main result shows that the edge-connectivity in all fiber graphs of this counterexample is best possible if we use moves from Graver basis instead.