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

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.

• The paper proves Tutte’s 3-flow conjecture for every connected simple Cayley graph of valency at least 4 on a solvable group of order 2n, where n is square-free.
• The authors characterize nowhere-zero Z₃-flows in 5-valent graphs using partitions into two pseudoforests, with transversals and perfect matchings covering the tree components when needed.
• The proof reduces potential counterexamples to four explicit Cayley graph families and verifies each constructively, while a 5-valent multigraph on A₄ shows that the simple-graph condition cannot be omitted.

Context and main result

Tutte's 3-flow conjecture asserts that every 4-edge-connected graph admits a nowhere-zero 3-flow (2603.24175). Since connected $k$-valent Cayley graphs are $k$-edge-connected (Mader), the conjecture restricted to Cayley graphs is equivalent to the statement that every connected Cayley graph of valency at least 4 admits a nowhere-zero 3-flow. Prior work had verified this for Cayley graphs on abelian, nilpotent, dihedral, generalized dihedral/quaternion/dicyclic groups, supersolvable groups with non-cyclic Sylow 2-subgroups, groups with square-free derived subgroup, and groups of order $p^2q$ or $8p$. The paper extends this program to a new infinite family of solvable groups.

The main theorem states that if $G$ is a solvable group of order $2n$ with $n$ square-free, then every connected Cayley graph on $G$ of valency at least 4 admits a nowhere-zero 3-flow. The result is proved only for simple Cayley graphs: the authors exhibit a 5-valent Cayley multigraph on $A_4$, namely $\mathrm{Cay}(A_4,\{a,b,b,b^{-1},b^{-1}\})$, that admits no nowhere-zero $k$0-flow. The argument is a clean contradiction: any such flow would yield a nowhere-zero 3-flow on a cubic graph obtained by smoothing degree-2 vertices, contradicting the classical fact that a cubic graph has a nowhere-zero 3-flow if and only if it is bipartite. This demonstrates that the theorem cannot be extended to multigraphs in this class.

A pseudoforest characterization of nowhere-zero 3-flows

The paper's principal methodological contribution is a necessary and sufficient condition for a 5-valent simple graph $k$1 to admit a nowhere-zero $k$2-flow, phrased purely combinatorially. It states that $k$3 admits such a flow if and only if $k$4 can be partitioned into non-empty sets $k$5 and $k$6 such that both induced subgraphs $k$7 and $k$8 are pseudoforests, and either all components of both are unicyclic, or there exist transversals $k$9 of $p^2q$0 and $p^2q$1 of $p^2q$2 (one vertex per tree component, none from unicyclic components) such that $p^2q$3 has a perfect matching.

The proof exploits the fact that a nowhere-zero $p^2q$4-flow may be taken constant on edges; at each vertex of valency 5, flow conservation forces the out-degree under the orientation to be 1 or 4 modulo 3. Setting $p^2q$5 as the out-degree-1 vertices and $p^2q$6 as the out-degree-4 vertices, restricting orientations appropriately yields $p^2q$7-orientations of $p^2q$8 and $p^2q$9, which exist exactly when these graphs are pseudoforests with suitable transversals. Conversely, given such a partition, orienting tree components toward their transversal vertices, cycles cyclically, matching edges from $8p$0 to $8p$1, and remaining cross-edges from $8p$2 to $8p$3 produces an orientation with all out-degrees in $8p$4, hence a constant-value nowhere-zero $8p$5-flow. This complements an earlier sufficient condition based on decompositions into circular ladders $8p$6 of odd order, which the authors also derive and use for one of the exceptional cases.

Structure of the proof of the main theorem

The proof proceeds by induction on $8p$7 and reduces to valency 5, since valency 4 gives Eulerian graphs trivially and valency at least 6 falls under the Lovász–Thomassen–Wu–Zhang theorem that every 6-edge-connected graph admits a nowhere-zero 3-flow. A structural reduction lemma shows that a hypothetical counterexample must be one of four explicitly described graphs:

Case	Group	Connection set
$8p$8	$8p$9	$G$0 or $G$1
$G$2	$G$3	same two forms

Here $G$4 is prime, $G$5, $G$6, $G$7 acts nontrivially on the normal Klein four-subgroup $G$8, and in the second family $G$9. The reduction relies on the Fitting subgroup analysis: since $2n$0, $2n$1 has a normal Sylow 2-subgroup of order 4 containing all involutions; the central involution in the connection set would immediately give a flow via Nănaşiová–Škoviera unless $2n$2, forcing $2n$3 acting on $2n$4. Quotient arguments then pin down the Fitting subgroup as $2n$5 with $2n$6 cyclic of prime order $2n$7, and the connection set takes one of the two displayed forms.

The verification that each of the four graphs admits a nowhere-zero $2n$8-flow splits according to the tools used. For $2n$9, the ladder-based sufficient condition applies: the subgraph generated by $n$0 decomposes into $n$1 circular ladders $n$2, and a carefully constructed $n$3-flow on the subgraph induced by $n$4 vanishes on at most one rung per ladder component. For $n$5, the authors define an explicit partition $n$6 (with $n$7 splitting $n$8 by parity of the exponent of $n$9) and verify directly that $G$0 consists of cycles, edges, and isolated vertices, hence is a pseudoforest; a transversal with perfect matching across the cut is constructed via an explicit adjacency map $G$1 satisfying $G$2, using the identity $G$3 established by a symmetry lemma showing $G$4.

For $G$5 and $G$6 (the case $G$7), the authors develop a general transfer mechanism: conditions on the cyclic subgraph $G$8 — namely that certain induced subgraphs $G$9 and $A_4$0 are pseudoforests whose tree components meet $A_4$1 in at most two vertices, together with compatible reduced transversals related by multiplication by $A_4$2, $A_4$3, $A_4$4, $A_4$5 — lift to a valid partition of the whole group. These conditions are then verified case by case depending on the residue of $A_4$6 modulo 4 and on the value of $A_4$7 where $A_4$8, including a special direct construction when $A_4$9 and explicit edge listings for $\mathrm{Cay}(A_4,\{a,b,b,b^{-1},b^{-1}\})$0 organized by the Euclidean division $\mathrm{Cay}(A_4,\{a,b,b,b^{-1},b^{-1}\})$1.

Limitations and open questions

The result is confined to simple Cayley graphs; the $\mathrm{Cay}(A_4,\{a,b,b,b^{-1},b^{-1}\})$2 multigraph example rules out a naive extension to multigraphs, though Lemma multi shows that multigraphs whose connection multiset contains an element of odd order greater than 2 with multiplicity 1 do admit flows. The structural reduction depends essentially on solvability through the Fitting subgroup machinery, and the paper leaves untouched whether Tutte's 3-flow conjecture holds for Cayley graphs on nonsolvable groups, or whether the square-free hypothesis on $\mathrm{Cay}(A_4,\{a,b,b,b^{-1},b^{-1}\})$3 can be relaxed. The pseudoforest criterion itself is stated only for valency 5; extending it to higher valencies remains open within this framework.

Conclusion

The paper verifies Tutte's 3-flow conjecture for Cayley graphs on solvable groups of twice square-free order, completing the argument through a combination of a new pseudoforest-decomposition characterization of nowhere-zero $\mathrm{Cay}(A_4,\{a,b,b,b^{-1},b^{-1}\})$4-flows in 5-valent graphs, a Fitting-subgroup-driven reduction to four explicit families of Cayley graphs, and detailed constructive verifications for those families. The pseudoforest criterion is a standalone contribution likely applicable to other classes of 5-valent graphs, while the explicit multigraph counterexample delineates precisely where the simple-graph hypothesis is necessary.