On removable edge subsets in graphs with a nowhere-zero $4$-flow
Davide Mattiolo
TL;DR
The paper addresses the problem of finding removable edge subsets in graphs that admit a nowhere-zero $4$-flow, proving that such graphs contain a $3$-removable set of size at most $|E(G)|/6$. The key technique is the use of flow-continuous maps, in particular a characterization that a graph has a $4$-NZF iff there exists a $\mathbb{Z}$-flow-continuous map to $K_4$, which yields a small preimage whose removal preserves a $3$-NZF. This leads to a $3$-flow with support at least $5/6$ of the edges and implies that bipartite cubic graphs satisfy Hoffmann-Ostenhof's conjecture, while providing a near-optimal approximation for the non-bipartite case; for $3$-edge-colorable cubic graphs, it guarantees a large spanning subgraph consisting of cycles or subdivisions of bipartite cubic graphs. Overall, the work connects flow theory and flow-continuous mappings to structural results on cubic and bipartite graphs, with implications for edge-colorings and related conjectures.
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. Rollová and R. Šámal [$3$-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 $|E(H)|\ge \frac{5}{6}|E(G)|$.