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

Abstract

Thomassen (J. Combin. Theory Ser. B 108 (2014), 81-91) showed that every graph admitting a nowhere-zero $3$-flow also admits an $S^1$-flow. He also proved that the converse holds for cubic graphs, but constructed counterexamples showing it fails in general. Wang et al. (SIAM J. Discrete Math. 29 (2015), 2166-2178) presented a couple of sufficient conditions under which the existence of an $S^1$-flow guarantees the existence of a nowhere-zero 3-flow. In this paper, we first prove that a graph with maximum degree at most four admits a nowhere-zero $3$-flow if and only if it admits an $S^1$-flow. We then develop some reduction techniques for $S^1$-flows based on graph operations including bull-growth, $2$-sums, and contractions. Finally, we apply those techniques to characterize triangularly connected graphs and graphs containing a spanning triangle-tree that admit $S^1$-flows, respectively.