Sublinear bounds for nullity of flows and approximating Tutte's flow conjectures

TL;DR

This paper establishes equivalences between Tutte's flow conjectures and the existence of sublinear bounds on the nullity of flows in certain classes of graphs, advancing understanding of flow properties and conjectures.

Contribution

It shows that Tutte's 3-, 4-, and 5-flow conjectures are equivalent to the existence of sublinear functions bounding nullity of flows in specific graph classes.

Findings

Tutte's 5-flow conjecture is equivalent to sublinear bounds on nullity in 3-edge-connected cubic graphs.

Tutte's 4-flow conjecture is equivalent to sublinear bounds on nullity in bridgeless graphs without Petersen minors.

Tutte's 3-flow conjecture is equivalent to sublinear bounds on nullity in 4-edge-connected graphs.

Abstract

A function $f:N\rightarrow N$ is sublinear, if \[\lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0.\] If $A$ is an Abelian group, $G$ is a graph and $\phi$ is an $A$-flow in $G$, then let $N(\phi)$ be the nullity of $\phi$, that is, the set of edges $e$ of $G$ with $\phi(e)=0$. In this paper we show that (a) Tutte's 5-flow conjecture is equivalent to the statement that there is a sublinear function $f$, such that all $3$-edge-connected cubic graphs admit a $\mathbb{Z}_5$-flow $\phi$ (not necessarily no-where zero), such that $|N(\phi)|\leq f(|E(G)|)$; (b) Tutte's 4-flow conjecture is equivalent to the statement that there is a sublinear function $f$, such that all bridgeless graphs without a Petersen minor admit a $\mathbb{Z}_4$-flow $\phi$ (not necessarily no-where zero), such that $|N(\phi)|\leq f(|E(G)|)$; (c) Tutte's 3-flow conjecture is equivalent to the statement that there is a sublinear function $f$, such that all $4$-edge-connected graphs admit a $\mathbb{Z}_3$-flow $\phi$ (not necessarily no-where zero), such that $|N(\phi)|\leq f(|E(G)|)$.