Nowhere-zero $3$-flows in Cayley graphs on solvable groups of twice square-free order

TL;DR

This paper proves Tutte's 3-flow conjecture for Cayley graphs on solvable groups of order twice a square-free number, introducing a new condition based on pseudoforest decomposition for 3-flows.

Contribution

It establishes the conjecture for a new class of graphs and provides a novel criterion involving pseudoforest decomposition for 3-flows.

Findings

Verification of Tutte's 3-flow conjecture for specified Cayley graphs

Development of a new necessary and sufficient condition for 3-flows in 5-valent graphs

Application of pseudoforest decomposition in flow analysis

Abstract

We verify Tutte's $3$-flow conjecture in the class of Cayley graphs on solvable groups of order $2n$, where $n$ is square-free. The proof relies on a new necessary and sufficient condition for a simple $5$-valent graph to admit a nowhere-zero $3$-flow in terms of a pseudoforest decomposition.