Tutte's $3$-Flow Conjecture in $3$-tree-connected graphs

TL;DR

This paper explores the extension of Tutte's 3-flow conjecture to 3-tree-connected graphs, linking it to the conjecture that 5-edge-connected graphs are Z_3-connected, and discusses implications for graph flow properties.

Contribution

It establishes that if the conjecture about 5-edge-connected graphs is true, then Tutte's 3-flow conjecture can be extended to 3-tree-connected graphs.

Findings

Shows the implication of the 5-edge-connected conjecture on 3-tree-connected graphs.

Proposes a potential extension of Tutte's 3-flow conjecture.

Connects two longstanding conjectures in graph theory.

Abstract

Tutte's $3$-flow conjecture says that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. Kochol (2001) showed that it is enough to prove this conjecture for $5$-edge-connected graphs. Former, Jaeger, Linial, Payan, and Tarsi (1992) conjectured that every $5$-edge-connected graph is $Z_3$-connected and so it admits a nowhere-zero $3$-flow. In this note, we show that if the second conjecture would be true, then every $3$-tree-connected graph must also be $Z_3$-connected and so Tutte's $3$-flow conjecture can be extended to this family of graphs.