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**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 \(\mathcal{G}\).

Reviewer: Van Bang Le (Rostock)

### MSC:

05C75 | Structural characterization of families of graphs |

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\textit{Z. Zhang} et al., J. Graph Theory 54, No. 4, 284--292 (2007; Zbl 1121.05099)

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### References:

[1] | and , Graph Theory With Applications, Elsevier Science Publishing Company, New York, 1976. · Zbl 1226.05083 · doi:10.1007/978-1-349-03521-2 |

[2] | and , Algebraic Graph Theory, Springer-Verlag Inc., New York, 2001. · doi:10.1007/978-1-4613-0163-9 |

[3] | Imrich, J Graph Theory 43 pp 93– (2003) |

[4] | Shu, J Graph Theory 50 pp 79– (2005) |

[5] | Tutte, Proc Lond Math Soc 51 pp 474– (1954) |

[6] | Tutte, J Canad Math Soc 6 pp 80– (1954) · Zbl 0055.17101 · doi:10.4153/CJM-1954-010-9 |

[7] | Introduction to Graph Theory (2nd edition), Prentice Hall, New York, 2001. |

[8] | Integer Flows and Cycle Covers of Graphs, Marcel Dekker Inc., New York, 1997. |

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