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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
##### Keywords:
circulant tournament; dichromatic number; vertex critical
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##### References:
 [1] Beineke, L.W.; Reid, K.B., Tournaments, (), 169-204 · Zbl 0434.05037 [2] Bondy, J.A.; Murty, U.S.R., Graph theory with applications, (1976), American Elsevier Pub. Co · Zbl 1134.05001 [3] Erdős, P.; Moser, L., On the representation of directed graphs as unions of orderings, Magyar tud. akad. mat. int. kutató int. Közl., 9, 125-132, (1964) · Zbl 0136.44901 [4] B. Llano, M. Olsen, On a conjecture of Víctor Neumann-Lara (submitted for publication) · Zbl 1341.05101 [5] Neumann-Lara, V., The dichromatic number of a digraph, J. combin. theory ser. B, 33, 265-270, (1982) · Zbl 0506.05031 [6] Neumann-Lara, V., The 3- and 4-dichromatic tournaments of minimum order, Discrete math., 135, 233-243, (1994) · Zbl 0829.05028 [7] Neumann-Lara, V., Note on vertex critical 4-dichromatic circulant tournaments, Discrete math., 170, 289-291, (1997) · Zbl 0876.05039 [8] Neumann-Lara, V., The acyclic disconnection of a digraph, Discrete math., 617-632, (1999), 197/198 · Zbl 0928.05033 [9] Neumann-Lara, V., Dichromatic number, circulant tournaments and zykov sums of digraphs, Discuss. math. graph theory, 2, 197-207, (2000) · Zbl 0984.05043 [10] Neumann-Lara, V.; Urrutia, J., Vertex critical r-dichromatic tournaments, Discrete math., 49, 83-87, (1994) · Zbl 0532.05031 [11] Reid, K.B.; Parker, E.T., Disproof of a conjecture of Erdős and Moser on tournaments, J. combin. theory, 9, 225-238, (1970) · Zbl 0204.24605 [12] Sanchez-Flores, A., On tournaments free of large transitive subtournaments, Graphs and combinatorics, 14, 181-200, (1998) · Zbl 0918.05058
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