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

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.