A minimal counterexample to a strengthening of Perles' conjecture

TL;DR

This paper provides a minimal counterexample to a strengthened version of Perles' conjecture, demonstrating a specific 4-polytope with unique graph properties that challenge previous assumptions.

Contribution

It introduces the first known minimal counterexample to a strengthened form of Perles' conjecture, clarifying the conjecture's limitations.

Findings

Identifies a 4-polytope with a special induced subgraph

Shows the subgraph is planar and not a facet graph

Provides a counterexample to the conjecture's strengthening

Abstract

In this paper, we present a minimal counterexample to a conjecture of Perles that answers a question of Haase and Ziegler. The example is a simple 4-polytope that has an induced 3-connected 3-regular subgraph, whose graph complement is connected. This subgraph is planar and not the graph of a facet of the polytope.