Group connectivity of graphs: a nonhomogeneous analogue of nowhere-zero flow properties
Journal of Combinatorial Theory Series B
Gro¨tzsch's 3-color theorem and its counterparts for the torus and the projective plane
Journal of Combinatorial Theory Series B
An equivalent version of the 3-flow conjecture
Journal of Combinatorial Theory Series B
Superposition and constructions of graphs without nowhere-zero k-flows
European Journal of Combinatorics
Graph Theory With Applications
Graph Theory With Applications
Nowhere-zero 3-flows in locally connected graphs
Journal of Graph Theory
Claw-decompositions and tutte-orientations
Journal of Graph Theory
Z3-connectivity of 4-edge-connected 2-triangular graphs
European Journal of Combinatorics
An extremal problem on group connectivity of graphs
European Journal of Combinatorics
Contractible configurations on 3-flows in graphs satisfying the Fan-condition
European Journal of Combinatorics
Nowhere-zero 3-flows and modulo k-orientations
Journal of Combinatorial Theory Series B
Hi-index | 0.00 |
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@?"2H"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"1T"2...T"m in G such that for 1=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"1T"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@?"2G"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 Barat and Thomassen that every triangulation of a surface admits all generalized Tutte-orientations.