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 byA 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 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.
For a prime p, let D4p be the dihedral group 〈a,b | a2p = b2 = 1, b-1ab = a-1〉 of order 4p, and Cay (G,S) a connected cubic Cayley graph of order 4p. In this paper, it is shown that the automorphism group Aut ( Cay (G,S)) of Cay (G,S) is the semiproduct R(G) ⋊ Aut (G,S), where R(G) is the right regular representation of G and Aut (G,S) = {α ∈ Aut (G) | Sα = S}, except either G = D4p (p ≥ 3), Sβ = {b,ab,apb} for some β ∈ Aut (D4p) and [Formula: see text], or Cay (G,S) is isomorphic to the three-dimensional hypercube Q3[Formula: see text] and G = ℤ4 × ℤ2 or D8.
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