Nowhere-zero flows in tensor product of graphs. (English) Zbl 1121.05099

Let \(k\) be a positive integer. A nowhere-zero \(k\)-flow of a graph \(G=(V,E)\) is an ordered pair \((D, f)\), where \(D\) is an orientation of \(G\) and \(f: E \rightarrow \{\pm 1, \pm 2\), \(\ldots, \pm(k-1)\}\) such that \(\sum_{e\in E^+(v)} f(e) = \sum_{e\in E^-(v)} f(e)\) holds for all vertices \(v\in V\), where \(E^+(v)\) and \(E^-(v)\) denote the set of outgoing and incoming edges incident with \(v\) with respect to \(D\), respectively. The tensor product of two graphs \(G_1 =(V_1, E_1)\) and \(G_2=(V_2, E_2)\) is the graph \(G_1\times G_2\) with vertex set \(V_1\times V_2\) and \((u_1, v_1)\) is adjacent to \((u_2, v_2)\) in \(G_1\times G_2\) if \(u_1\) is adjacent to \(u_2\) in \(G_1\) and \(v_1\) is adjacent to \(v_2\) in \(G_2\). The main result of the paper is: If the minimum degree of \(G_1\) is at least two and if \(G_2\) does not belong to a certain graph class \(\mathcal{G}\), then \(G_1\times G_2\) has a nowhere-zero \(3\)-flow. The graph class \(\mathcal{G}\) is defined as follows: The 2-vertex complete graph is a member of \(\mathcal{G}\) and for any two members \(H_1, H_2\) of \(\mathcal{G}\), the graph obtained from \(H_1\) and \(H_2\) by joining \(H_1\) and \(H_2\) with an edge is also a member of \(\mathcal{G}\).


05C75 Structural characterization of families of graphs
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