On removable edge subsets in graphs with a nowhere-zero $4$-flow

TL;DR

This paper proves that graphs with a nowhere-zero 4-flow contain large subsets of edges whose removal results in a graph with a nowhere-zero 3-flow, advancing understanding of flow properties and edge removability.

Contribution

It establishes the existence of large 3-removable edge subsets in graphs with a 4-flow, confirming a conjecture for such graphs and providing bounds related to Hoffmann-Ostenhof's conjecture.

Findings

Every graph with a nowhere-zero 4-flow has a 3-removable subset of at most 1/6 of edges.

Bipartite cubic graphs with a 4-flow satisfy Hoffmann-Ostenhof's conjecture.

In 3-edge-colorable cubic graphs, a large subgraph with specific properties contains at least 5/6 of the edges.

Abstract

A set $R\subseteq E(G)$ of a graph $G$ is $k$-removable if $G-R$ has a nowhere-zero $k$-flow. We prove that every graph $G$ admitting a nowhere-zero $4$-flow has a $3$-removable subset consisting of at most $\frac{1}{6}|E(G)|$ edges. This gives a positive answer to a conjecture of M. DeVos, J. McDonald, I. Pivotto, E. Rollov\'a and R. \v{S}\'amal [$3$-Flows with large support, J. Comb. Theory Ser. B 144 (2020), 32-80] in the case of graphs admitting a nowhere-zero $4$-flow. Moreover, Hoffmann-Ostenhof recently conjectured that every cubic graph with a nowhere-zero $4$-flow has a $4$-removable edge. Bipartite cubic graphs verify this conjecture. Our result gives an approximation for Hoffmann-Ostenhof's Conjecture in the non-bipartite case. Finally, for cubic graphs, our result implies that every $3$-edge-colorable cubic graph $G$ contains a subgraph $H$ whose connected components are either cycles or subdivisions of bipartite cubic graphs, such that $|E(H)|\ge \frac{5}{6}|E(G)|$.