Nowhere-zero 5-flows on cubic graphs with oddness 4

TL;DR

This paper proves that all cyclically 6-edge-connected cubic graphs with oddness at most 4 admit a nowhere-zero 5-flow, advancing understanding of Tutte's 5-Flow Conjecture in specific graph classes.

Contribution

It establishes the existence of nowhere-zero 5-flows for a new class of cubic graphs with bounded oddness and cyclic connectivity.

Findings

Cubic graphs with oddness ≤ 4 and cyclically 6-edge-connected have nowhere-zero 5-flows.

Supports the broader conjecture for specific graph classes.

Provides new insights into Tutte's 5-Flow Conjecture.

Abstract

Tutte's 5-Flow Conjecture from 1954 states that every bridgeless graph has a nowhere-zero 5-flow. In 2004, Kochol proved that the conjecture is equivalent to its restriction on cyclically 6-edge connected cubic graphs. We prove that every cyclically 6-edge-connected cubic graph with oddness at most 4 has a nowhere-zero 5-flow.