Nowhere-zero 3-flows in triangularly connected graphs

Let H1 and H2 be two subgraphs of a graph G. We say that G is the 2-sum of H1 and H2, denoted by H1 ⊕2 H2, if E(H1)∪E(H2)=E(G), |V (H1)∩ V (H2)| = 2, and |E(H1)∩E(H2)| = 1. A triangle-path in a graph G is a sequence of distinct triangles T1T2 · · ·Tm in G such that for 1 i m − 1, |E(Ti) ∩ E(Ti+1)| = 1 and E(Ti) ∩ E(Tj ) = ∅ if j > i + 1. A connected graph G is triangularly connected if for any two edges e and e′, which are not parallel, there is a triangle-path T1T2 · · ·Tm such that e ∈E(T1) and e′ ∈ E(Tm). Let G be a triangularly connected graph with at least three vertices. We prove that G has no nowhere-zero 3-flow if and only if there is an odd wheel W and a subgraph G1 such that G=W ⊕2 G1, where G1 is a triangularly connected graph without nowhere-zero 3-flow. Repeatedly applying the result, we have a complete characterization of triangularly connected graphs which have no nowhere-zero 3-flow. As a consequence, G has a nowhere-zero 3-flow if it contains at most three 3-cuts. This verifies Tutte’s 3-flow conjecture and an equivalent version by Kochol for triangularly connected graphs. By the characterization, we obtain extensions to earlier results on locally connected graphs, chordal graphs and squares of graphs. As a corollary, we obtain a result of Barát and Thomassen that every triangulation of a surface admits all generalized Tutte-orientations. © 2008 Elsevier Inc. All rights reserved.