The Strengthened Hanna Neumann Conjecture I: A Combinatorial Proof

TL;DR

This paper provides a concise combinatorial proof of the Strengthened Hanna Neumann Conjecture, simplifying previous cohomological methods and building on prior results for the rank two case.

Contribution

It introduces a new combinatorial proof of the conjecture, replacing complex cohomological techniques with an inductive approach.

Findings

Proof confirms the conjecture in its graph theoretic form

Simplifies the proof technique from cohomology to combinatorics

Builds on Tardos' result for the rank two case

Abstract

We prove the Strengthened Hanna Neumann Conjecture, in its common graph theoretic formulation. Our original approach to this conjecture used cohomology of sheaves on graphs, although here we give a short combinatorial proof that we found in a succession of simplifications of the cohomological approach. Our proof is inductive, and requires Tardos' previous result settling the rank two case of the conjecture.