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

英文摘要：

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 $ϕ$ is an $A$-flow in $G$, then let $N(ϕ)$ be the nullity of $ϕ$, that is, the set of edges $e$ of $G$ with $ϕ(e)=0$. 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 $ϕ$ (not necessarily no-where zero), such that $|N(ϕ)|\leq f(|E(G)|)$; (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 $ϕ$ (not necessarily no-where zero), such that $|N(ϕ)|\leq f(|E(G)|)$; (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 $ϕ$ (not necessarily no-where zero), such that $|N(ϕ)|\leq f(|E(G)|)$.