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

08/17/2020 ∙ by Vahan Mkrtchyan, et al. ∙ 0

A function f:N→ N is sublinear, if lim_x→ +∞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 ℤ_5-flow ϕ (not necessarily no-where zero), such that |N(ϕ)|≤ 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 ℤ_4-flow ϕ (not necessarily no-where zero), such that |N(ϕ)|≤ 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 ℤ_3-flow ϕ (not necessarily no-where zero), such that |N(ϕ)|≤ f(|E(G)|).

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