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

Abstract

While every graph admitting a nowhere-zero $3$-flow also admits an $S^1$-flow, the converse does not hold in general as shown by Thomassen (2014). In this paper, we develop reduction techniques for $S^1$-flows based on graph operations including bull-growth, $2$-sums, and wheel contractions. A key tool is the two-terminal $S^1$-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 $S^1$-flow. As applications, we completely characterize $S^1$-flows in two graph classes: a triangularly connected graph admits an $S^1$-flow if and only if it is not an odd wheel; and a graph containing a spanning triangle-tree admits an $S^1$-flow if and only if it is not an odd crystal.