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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
##### Keywords:
Edge coloured tournament; Monochromatic directed path
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##### References:
 [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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