Nowhere-zero 3-flows in triangularly connected graphs

Let H 1 and H 2 be two subgraphs of a graph G. We say that G is the 2-sum of H 1 and H 2 , denoted by H 1 ⊕ 2 H 2 , if E ( H 1 ) ∪ E ( H 2 ) = E ( G ) , | V ( H 1 ) ∩ V ( H 2 ) | = 2 , and | E ( H 1 ) ∩ E ( H 2 ) | = 1 . A triangle-path in a graph G is a sequence of distinct triangles T 1 T 2 ⋯ T m in G such that for 1 ⩽ i ⩽ m − 1 , | E ( T i ) ∩ E ( T i + 1 ) | = 1 and E ( T i ) ∩ E ( T j ) = ∅ 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 T 1 T 2 ⋯ T m such that e ∈ E ( T 1 ) and e ′ ∈ E ( T m ) . 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 G 1 such that G = W ⊕ 2 G 1 , where G 1 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 Tutteʹs 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.