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A conjecture of Neumann-Lara on infinite families of \(r\)-dichromatic circulant tournaments. (English) Zbl 1185.05071
Summary: We exhibit infinite families of vertex critical \(r\)-dichromatic circulant tournaments for all \(r \geq 3\). The existence of these infinite families was conjectured by Neumann-Lara [V. Neumann-Lara, “Vertex critical 4-dichromatic circulant tournaments”, Discrete Math. 170, No. 1-3, 289–291 (1997; Zbl 0876.05039)], who later proved it for all \(r \geq 3\) and \(r \neq 7\). Using different methods, we provide new constructions of such infinite families for all \(r \geq 3\), which covers the case \(r = 7\) and thus settles the conjecture.

MSC:
05C20 Directed graphs (digraphs), tournaments
05C15 Coloring of graphs and hypergraphs
05C63 Infinite graphs
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