Nowhere-zero 3-flows in triangularly connected graphs. (English) Zbl 1171.05026

Summary: 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 \(H_1\oplus _2 H_2\), if \(E(H_1)\cup E(H_2)=E(G)\), \(|V(H_1)\cap V(H_2)|=2\), and \(|E(H_1)-E(H_2)|=1\). A triangle-path in a graph \(G\) is a sequence of distinct triangles \(T_1T_2\dots T_m\) in \(G\) such that for \(1\leq i\leq m-1\), \(|E(T_i)-E(T_{i+1})|=1\) and \(E(T_i)\cap E(T_j)=\emptyset\) 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 \(T_1T_2\dots T_m\) such that \(e\in E(T_1)\) and \(e'\in E(T_m)\). 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 \(G=W\oplus_2G_1\), 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 J. Barát and C. Thomassen [J. Graph Theory 52, No. 2, 135–146 (2006; Zbl 1117.05088)] that every triangulation of a surface admits all generalized Tutte-orientations.


05C20 Directed graphs (digraphs), tournaments
05C40 Connectivity


Zbl 1117.05088
Full Text: DOI


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