Group connectivity of graphs — a nonhomogeneous analogue of nowhere-zero flow properties. (English) Zbl 0824.05043

Summary: Let \(G= (V, E)\) be a digraph and \(f\) a mapping from \(E\) into an Abelian group \(A\). Associated with \(f\) is its boundary \(\partial f\), a mapping from \(V\) to \(A\), defined by \(\partial f(x)= \sum_{e\text{ leaving }x} f(e)- \sum_{e\text{ entering }x} f(e)\). We say that \(G\) is \(A\)- connected if for every \(b: V\to A\) with \(\sum_{x\in V} b(x)= 0\) there is an \(f: E\to A -\{0\}\) with \(b= \partial f\). This concept is closely related to the theory of nowhere-zero flows and is being studied here in the light of that theory.


05C40 Connectivity
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C20 Directed graphs (digraphs), tournaments
05C15 Coloring of graphs and hypergraphs
05B35 Combinatorial aspects of matroids and geometric lattices
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