Reduction Operations and Structural Characterizations of $S^1$-Flows in Graphs

Abstract: While every graph admitting a nowhere-zero $3$-flow also admits an $S<sup>1$-flow, the converse does not hold in general as shown by Thomassen (2014). In this paper, we develop reduction techniques for $S<sup>1$-flows based on graph operations including bull-growth, $2$-sums, and wheel contractions. A key tool is the two-terminal $S<sup>1$-preflow, which enables us to prove that if a $2$-connected graph contains an odd wheel as a proper subgraph and contracting the wheel yields a graph with a nowhere-zero $3$-flow, then the original graph admits an $S<sup>1$-flow. As applications, we completely characterize $S<sup>1$-flows in two graph classes: a triangularly connected graph admits an $S<sup>1$-flow if and only if it is not an odd wheel; and a graph containing a spanning triangle-tree admits an $S<sup>1$-flow if and only if it is not an odd crystal.

• The paper develops bull-growth, generalized 2-sum, wheel-contraction, and preflow techniques that preserve or construct S¹-flows under structural reductions.
• Bridgeless triangularly connected graphs admit an S¹-flow exactly when they are not odd wheels, covering locally connected, squared, and 2-connected chordal graph classes.
• Graphs with a spanning triangle-tree admit an S¹-flow exactly when they are not odd crystals, while the paper conjectures that every S¹-flow graph has a nowhere-zero 4-flow.

Context and motivation

Integer flow theory, introduced by Tutte as the dual of vertex coloring, has been extended to real-valued flows by Goddyn, Tarsi, and Zhang, and further to vector flows by Jain and Thomassen. This paper, by Li, Li, Luo, and Su (2608.18725), concerns $S^1$-flows: assignments of unit vectors in $\mathbb{R}^2$ to edges satisfying the conservation condition $\partial f(v)=\mathbf{0}$ at every vertex. Thomassen established that every graph with a nowhere-zero $3$-flow admits an $S^1$-flow (via an $R_3$-flow of cube roots of unity), that the converse holds for cubic graphs if and only if the graph is bipartite, but fails in general: graphs built by repeated $2$-sums of copies of $K_4$ admit $S^1$-flows yet no nowhere-zero $3$-flow. Wang et al. posed the problem of characterizing conditions under which an $\mathbb{R}^2$0-flow forces a nowhere-zero $\mathbb{R}^2$1-flow, obtaining partial results via rank conditions on the flow and degree-$\mathbb{R}^2$2 structure. The present paper attacks this problem from the structural side, developing reduction operations—bull-growth, generalized $\mathbb{R}^2$3-sums, and wheel contractions—and using them to obtain complete characterizations of $\mathbb{R}^2$4-flows in two graph classes.

Reduction operations

The first reduction shows that the bull-growth operation preserves $\mathbb{R}^2$5-flows. Given a graph $\mathbb{R}^2$6, the bull-growth replaces an edge $\mathbb{R}^2$7 (or a non-edge when $\mathbb{R}^2$8) by two adjacent $\mathbb{R}^2$9-vertices $\partial f(v)=\mathbf{0}$0 sharing a common neighbor $\partial f(v)=\mathbf{0}$1. The proof in the case $\partial f(v)=\mathbf{0}$2 is a direct construction: orient all edges into $\partial f(v)=\mathbf{0}$3 and out of $\partial f(v)=\mathbf{0}$4, copy the value $\partial f(v)=\mathbf{0}$5 onto $\partial f(v)=\mathbf{0}$6 and $\partial f(v)=\mathbf{0}$7, and use the fact that any unit vector extends to a triple of unit vectors summing to zero to fill the remaining three edges. Consequently, if $\partial f(v)=\mathbf{0}$8 admits an $\partial f(v)=\mathbf{0}$9-flow then so does its bull-growth; this is used later in contrapositive form, since it implies bull-reductions of counterexamples remain counterexamples.

For $3$0-sums, the paper proves a closure theorem stronger than mere preservation: if $3$1 and both factors admit $3$2-flows, then replacing $3$3 by any number $3$4 of parallel edges still yields an $3$5-flow. The argument rotates one factor's flow so that the values on the common edge are either opposite ($3$6 even) or separated by $3$7 ($3$8 odd), then distributes their sum across the parallel edges. The mixed case—where one factor has only a nowhere-zero $3$9-flow after contracting $S^1$0—is handled by two complementary tools:

The two-terminal preflow lemma: if $S^1$1 has a nowhere-zero $S^1$2-flow, then for any distinct vertices $S^1$3 and any prescribed unit vector $S^1$4, there exists an $S^1$5-preflow on $S^1$6 whose boundary is $S^1$7 at $S^1$8, $S^1$9 at $R_3$0, and zero elsewhere. The construction modifies a $R_3$1-valued $R_3$2-flow along a directed $R_3$3–$R_3$4 path, extracts an Euler-trail parity decomposition, and encodes the result geometrically using the vertices of an equilateral triangle, so that each edge receives a vector of norm exactly $R_3$5.

A boundary lemma for odd wheels: every nonzero zero-sum boundary $R_3$6 on an odd wheel can be realized by a nowhere-zero $R_3$7-flow, via a sign-change recursion along the rim. A companion lemma gives odd wheels the analogous preflow property with arbitrary boundary vectors of norm $R_3$8, constructed explicitly in the complex plane using sixth roots of unity.

Combining these yields the main reduction theorem for $R_3$9-sums: if each factor either admits an $2$0-flow or has a contraction $2$1 admitting a nowhere-zero $2$2-flow, then the sum admits an $2$3-flow. In the hardest subcase (both factors have only contractive $2$4-flows), the preflow lemma supplies prefows on $2$5 and $2$6 whose boundary directions are chosen so that the angle between them satisfies $2$7, making the compensating value on $2$8 itself a unit vector.

Finally, the wheel-contraction theorem states that if a $2$9-connected graph contains a wheel $K_4$0 as a proper subgraph and $K_4$1 admits a nowhere-zero $K_4$2-flow, then $K_4$3 admits an $K_4$4-flow. If $K_4$5 is even or non-induced, $K_4$6 contains a $K_4$7-connected subgraph (using that $K_4$8 and even wheels are $K_4$9-connected), so $S^1$0 actually has a nowhere-zero $S^1$1-flow. For induced odd wheels, the boundary induced by the flow on $S^1$2 is either nonzero—in which case the boundary lemma extends it to a genuine $S^1$3-flow—or identically zero, in which case a component of $S^1$4 meeting the wheel in at least two vertices (guaranteed by $S^1$5-connectivity) is matched against the wheel via the two preflow lemmas with a common boundary vector.

Triangularly connected graphs

A graph is triangularly connected if every pair of edges lies on a common triangle-path. Fan et al. characterized such graphs without nowhere-zero $S^1$6-flows as those reducible by $S^1$7-sums to odd wheels. Building on this, the paper proves:

Theorem. A bridgeless triangularly connected graph admits an $S^1$8-flow if and only if it is not an odd wheel.

The necessity direction uses Wang et al.'s criterion: in an odd wheel, the degree-$S^1$9 vertices induce a connected cycle while removing them leaves an acyclic graph, and odd wheels have no $3$0-flow. For sufficiency, a minimal counterexample must decompose as a $3$1-sum $3$2 with $3$3 an odd wheel and $3$4 triangularly connected without a $3$5-flow; minimality forces $3$6 to be an odd wheel, and since $3$7 always admits a $3$8-flow (proved via planar duality and explicit colorings), the general $3$9-sum theorem produces an $\mathbb{R}^2$00-flow—a contradiction. Because locally connected graphs, squares of connected graphs, and $\mathbb{R}^2$01-connected chordal graphs are triangularly connected, the result immediately applies to these classes: every bridgeless member other than an odd wheel admits an $\mathbb{R}^2$02-flow.

Graphs with spanning triangle-trees

A triangle-tree is built from a triangle by repeatedly adding a vertex adjacent to exactly two adjacent existing vertices; a crystal is a triangle-path plus an edge joining its two leaves, and is odd if all degrees are odd. The second main characterization reads:

Theorem. A bridgeless graph containing a spanning triangle-tree admits an $\mathbb{R}^2$03-flow if and only if it is not an odd crystal.

The exclusion of odd crystals rests on a rigidity argument. At any $\mathbb{R}^2$04-vertex carrying an $\mathbb{R}^2$05-flow, the incident values (up to sign) form a regular hexagon on the unit circle. An algebraic lemma over the Eisenstein integers $\mathbb{R}^2$06 then shows that if several hexagon vertices plus two additional unit vectors sum to zero with the latter not canceling, the two extra vectors must also be hexagon vertices—the key cases being squared norms $\mathbb{R}^2$07, with $\mathbb{R}^2$08 excluded by the congruence $\mathbb{R}^2$09. Using the canonical outerplane embedding of a triangle-path (whose backbone is a path with pendant degree-$\mathbb{R}^2$10 vertices), a minimal-counterexample argument propagates hexagon membership from one leaf across the backbone; the propagation must break at some backbone vertex $\mathbb{R}^2$11, but the lemma forces the two exceptional edges back into the hexagon unless they cancel—which would allow a splitting operation producing a smaller odd crystal with an $\mathbb{R}^2$12-flow, contradicting minimality. Hence no odd crystal admits an $\mathbb{R}^2$13-flow.

For the converse, the paper invokes Li, Li, and Wang's structure theorem: a graph with a spanning triangle-tree and no $\mathbb{R}^2$14-flow is either $\mathbb{R}^2$15 or a bull-growth of a smaller such graph. Minimality reduces to the case where the smaller graph is an odd crystal, and a combinatorial lemma shows that bull-growing an odd crystal either destroys the spanning triangle-tree property, yields another odd crystal, or produces an $\mathbb{R}^2$16-flow (via a $\mathbb{R}^2$17-sum with $\mathbb{R}^2$18 or with a wheel)—each alternative contradicting the choice of counterexample.

Notably, the two classes characterized are incomparable: the paper exhibits a triangularly connected graph (a wheel with rim-pendant triangles) containing no spanning triangle-tree, and a crystal that is not triangularly connected because its added edge lies in no triangle.

Limitations and open questions

The paper concedes that its reduction techniques do not settle the general relationship between $\mathbb{R}^2$19-flows and integer flows. All known classes of graphs with $\mathbb{R}^2$20-flows also admit nowhere-zero $\mathbb{R}^2$21-flows, motivating the conjecture that every $\mathbb{R}^2$22-flow graph admits a nowhere-zero $\mathbb{R}^2$23-flow; the authors note that a minimal counterexample would have to be non-planar, non-cubic, and essentially $\mathbb{R}^2$24-edge-connected, but state plainly that they have neither proof nor disproof. They also leave open whether the wheel in the contraction theorem can be replaced by arbitrary one-edge-deletions of $\mathbb{R}^2$25-connected graphs, observing that $\mathbb{R}^2$26 shows some finite exceptions are unavoidable. Similar contraction results for graph classes beyond wheels are deferred to future work. The sufficiency proofs also rely structurally on prior characterizations (Fan et al.; Li–Li–Wang), so the method's reach is bounded by the availability of analogous structure theorems for other classes.

Conclusion

This paper develops a coherent toolkit—bull-growth preservation, parallel-edge closure under $\mathbb{R}^2$27-sums, two-terminal $\mathbb{R}^2$28-preflows with $\mathbb{R}^2$29-norm boundaries, and boundary-realization lemmas for odd wheels—for lifting flows from reduced graphs to larger ones. Its main results give exact structural obstructions to $\mathbb{R}^2$30-flows in two natural classes: odd wheels among triangularly connected graphs, and odd crystals among graphs with spanning triangle-trees. In both cases the obstruction coincides with the absence of a nowhere-zero $\mathbb{R}^2$31-flow, providing new instances where the $\mathbb{R}^2$32-flow and $\mathbb{R}^2$33-flow existence questions align, while the authors' own conjecture on $\mathbb{R}^2$34-flows marks precisely where current understanding ends.