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A counterexample to a conjecture on edge-coloured tournaments. (English) Zbl 1042.05039
Summary: We call the tournament \(T\) an \(m\)-coloured tournament if the arcs of \(T\) are coloured with \(m\) colours. In this paper we prove that for each \(n\geqslant 6\), there exists a 4-coloured tournament \(T_n\) of order \(n\) satisfying the two following conditions: (1) \(T_n\) does not contain \(C_3\) (the directed cycle of length 3, whose arcs are coloured with three distinct colours), and (2) \(T_n\) does not contain any vertex \(v\) such that for every other vertex \(x\) of \(T_n\), there is a monochromatic directed path from \(x\) to \(v\). This answers a question proposed by S. Minggang [J. Comb. Theory, Ser. B 45, 108–111 (1988; Zbl 0654.05033)].

MSC:
05C15 Coloring of graphs and hypergraphs
05C20 Directed graphs (digraphs), tournaments
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[1] Minggang, S., On monochromatic paths in m-coloured tournaments, J. combin. theory ser. B, 45, 108-111, (1988) · Zbl 0654.05033
[2] B. Sands, N. Sauer, R. Woodrow, On monochromatic paths in edge-coloured digraphs, J. Combin. Theory Ser. B (1982) 271-275. · Zbl 0488.05036
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