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nowhere-zero $\lambda $-flow; minimum nowhere-zero flow number; join of two graphs
Let $G$ be a graph, and $\lambda $ the smallest integer for which $G$ has a nowhere-zero $\lambda $-flow, i.e., an integer $\lambda $ for which $G$ admits a nowhere-zero $\lambda $-flow, but it does not admit a $(\lambda -1)$-flow. We denote the minimum flow number of $G$ by $\Lambda (G)$. In this paper we show that if $G$ and $H$ are two arbitrary graphs and $G$ has no isolated vertex, then $\Lambda (G \vee H) \leq 3$ except two cases: (i) One of the graphs $G$ and $H$ is $K_2$ and the other is $1$-regular. (ii) $H = K_1$ and $G$ is a graph with at least one isolated vertex or a component whose every block is an odd cycle. Among other results, we prove that for every two graphs $G$ and $H$ with at least $4$ vertices, $\Lambda (G \vee H) \leq 3$.
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