## 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 $$\mathcal{G}$$.

### MSC:

 05C75 Structural characterization of families of graphs

### Keywords:

Nowhere-zero flow; tensor product
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### References:

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