A counterexample to a conjecture on edge-coloured tournaments.

*(English)*Zbl 1042.05039Summary: 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)].

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\textit{H. Galeana-Sánchez} and \textit{R. Rojas-Monroy}, Discrete Math. 282, No. 1--3, 275--276 (2004; Zbl 1042.05039)

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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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