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.

💡 Research Summary

The paper addresses Tutte’s long‑standing 3‑flow conjecture, which asserts that every 4‑edge‑connected graph admits a nowhere‑zero 3‑flow, by focusing on a specific family of Cayley graphs. The authors consider finite solvable groups G of order 2n where n is square‑free, and they prove that every connected Cayley graph on such a group with valency at least four possesses a nowhere‑zero 3‑flow. This result extends earlier work that covered abelian, nilpotent, dihedral, generalized dihedral, quaternion, dicyclic, and several other families of groups.

The paper is organized as follows. Section 2 gathers necessary preliminaries from graph theory and group theory. It defines A‑flows, nowhere‑zero flows, and parity subgraphs, and recalls that a parity subgraph admitting a k‑flow lifts to a k‑flow on the whole graph (Proposition 2.1). The authors introduce Cayley (multi)graphs, the notion of a connection multiset, and the operation of taking quotients with respect to a normal subgroup. Important known results are restated: closed ladders (circular and Möbius ladders) admit a nowhere‑zero 3‑flow exactly when their length is even (Proposition 2.3); Cayley graphs containing a central involution admit a nowhere‑zero 3‑flow (Proposition 2.4); all Cayley graphs of valency at least four on nilpotent groups have such a flow (Proposition 2.5); and groups whose derived subgroup is square‑free also satisfy the conjecture (Proposition 2.6). Lemma 2.9 provides a new sufficient condition for a simple 5‑regular graph to have a nowhere‑zero Z₃‑flow, based on the presence of a spanning collection of odd‑length circular ladders together with a carefully chosen auxiliary subgraph Λ.

Section 3 introduces the central technical innovation: a decomposition of a 5‑regular graph into two pseudoforests. Lemma 3.1 shows that a (0, 1)‑orientation (each vertex has out‑degree 0 or 1) exists if and only if the graph is a pseudoforest, and that a prescribed transversal of the tree components can be realized as the set of vertices with out‑degree 0. Lemma 3.2 then proves a necessary and sufficient condition: a 5‑regular graph admits a nowhere‑zero Z₃‑flow precisely when its vertex set can be partitioned into non‑empty subsets U and W such that (i) the induced subgraphs Γ