Geodesic
Join of two graphs admits a nowhere-zero $3$-flow
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 433-446

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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, $\Lambda (G \vee H) \leq 3$.
DOI : 10.1007/s10587-014-0110-0
Classification : 05C20, 05C21, 05C78
Keywords: nowhere-zero $\lambda $-flow; minimum nowhere-zero flow number; join of two graphs 
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     title = {Join of two graphs admits a nowhere-zero $3$-flow},
     journal = {Czechoslovak Mathematical Journal},
     pages = {433--446},
     publisher = {mathdoc},
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Akbari, Saieed; Aliakbarpour, Maryam; Ghanbari, Naryam; Nategh, Emisa; Shahmohamad, Hossein. Join of two graphs admits a nowhere-zero $3$-flow. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 433-446. doi: 10.1007/s10587-014-0110-0

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