Let G be a graph, and λ the smallest integer for which G has a nowherezero λ-flow, i.e., an integer λ for which G admits a nowhere-zero λ-flow, but it does not admit a (λ − 1)-flow. We denote the minimum flow number of G by Λ(G). In this paper we show that if G and H are two arbitrary graphs and G has no isolated vertex, then Λ(G ∨ H) ⩽ 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, Λ(G ∨ H) ⩽ 3.
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Akbari, S., Aliakbarpour, M., Ghanbari, N. et al. Join of two graphs admits a nowhere-zero 3-flow. Czech Math J 64, 433–446 (2014). https://doi.org/10.1007/s10587-014-0110-0