A Simplicial Tutte "5"-flow Conjecture

TL;DR

This paper extends Tutte's 5-flow conjecture from graphs to higher-dimensional simplicial complexes, establishing bounds on the necessary flow value based on the complex's dimension.

Contribution

It generalizes the concept of nowhere-zero flows to simplicial complexes and provides bounds on the flow value needed for all bridgeless complexes of a given dimension.

Findings

Established a lower bound linear in the dimension d

Provided a partial exponential upper bound in d

Connected flow invariants to simplicial complex properties

Abstract

This paper concerns a generalization of nowhere-zero modular q-flows from graphs to simplicial complexes of dimension d greater than 1. A modular q-flow of a simplicial complex is an element of the kernel of the d-th boundary map with coefficients in Z/qZ; it is called nowhere-zero if it is not zero restricted to any of the facets of the complex. Briefly noting connections to other invariants of simplicial complexes, this paper provides a generalization of Tutte's 5-flow conjecture, which claims the universal existence of a 5-flow for all bridgeless graphs. Once phrased, this paper concludes with bounds on what "5" ought to be for simplicial complexes of dimension d: proving a lower bound linear in d and a partial upper bound exponential in d.