A note on nowhere-zero 3-flow and Z_3-connectivity

TL;DR

This paper advances the understanding of nowhere-zero 3-flows and Z_3-connectivity by proving new conditions under which graphs admit such flows, contributing partial results to longstanding conjectures in integer flow theory.

Contribution

The paper presents new partial results on 3-flow and Z_3-connectivity conjectures, specifically for graphs with certain edge-cut constraints, extending previous connectivity bounds.

Findings

Graphs with certain edge-cut limitations admit nowhere-zero 3-flows.

Bridgeless graphs with no 5-edge-cuts and limited 3-edge-cuts are shown to have 3-flows.

5-edge-connected graphs with bounded 5-edge-cuts are Z_3-connected.

Abstract

There are many major open problems in integer flow theory, such as Tutte's 3-flow conjecture that every 4-edge-connected graph admits a nowhere-zero 3-flow, Jaeger et al.'s conjecture that every 5-edge-connected graph is $Z_3$-connected and Kochol's conjecture that every bridgeless graph with at most three 3-edge-cuts admits a nowhere-zero 3-flow (an equivalent version of 3-flow conjecture). Thomassen proved that every 8-edge-connected graph is $Z_3$-connected and therefore admits a nowhere-zero 3-flow. Furthermore, Lov$\acute{a}$sz, Thomassen, Wu and Zhang improved Thomassen's result to 6-edge-connected graphs. In this paper, we prove that: (1) Every 4-edge-connected graph with at most seven 5-edge-cuts admits a nowhere-zero 3-flow. (2) Every bridgeless graph containing no 5-edge-cuts but at most three 3-edge-cuts admits a nowhere-zero 3-flow. (3) Every 5-edge-connected graph with at most five 5-edge-cuts is $Z_3$-connected. Our main theorems are partial results to Tutte's 3-flow conjecture, Kochol's conjecture and Jaeger et al.'s conjecture, respectively.