A counterexample to a conjecture on edge-coloured tournaments
نویسندگان
چکیده
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¿ 6, there exists a 4-coloured tournament Tn of order n satisfying the two following conditions: (1) Tn does not contain C3 (the directed cycle of length 3, whose arcs are coloured with three distinct colours), and (2) Tn does not contain any vertex v such that for every other vertex x of Tn, there is a monochromatic directed path from x to v. This answers a question proposed by Shen Minggang in 1988. c © 2004 Elsevier B.V. All rights reserved. MSC: 05C20
منابع مشابه
On the oriented perfect path double cover conjecture
An oriented perfect path double cover (OPPDC) of a graph $G$ is a collection of directed paths in the symmetric orientation $G_s$ of $G$ such that each arc of $G_s$ lies in exactly one of the paths and each vertex of $G$ appears just once as a beginning and just once as an end of a path. Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete Math. 276 (2004) 287-294) conjectured that ...
متن کاملChromatic-index-critical graphs of orders 13 and 14
A graph is chromatic-index-critical if it cannot be edge-coloured with ∆ colours (with ∆ the maximal degree of the graph), and if the removal of any edge decreases its chromatic index. The Critical Graph Conjecture stated that any such graph has odd order. It has been proved false and the smallest known counterexample has order 18 [18, 31]. In this paper we show that there are no chromatic-inde...
متن کاملKing-Serf Duo by Monochromatic Paths in k-Edge-Coloured Tournaments
An open conjecture of Erdős states that for every positive integer k there is a (least) positive integer f(k) so that whenever a tournament has its edges colored with k colors, there exists a set S of at most f(k) vertices so that every vertex has a monochromatic path to some point in S. We consider a related question and show that for every (finite or infinite) cardinal κ > 0 there is a cardin...
متن کاملEdge-disjoint Hamiltonian Paths and Cycles in Tournaments
We describe sufficient conditions for the existence of Hamiltonian paths in oriented graphs and use these to provide a complete description of the tournaments with no two edge-disjoint Hamiltonian paths. We prove that tournaments with small irregularity have many edge-disjoint Hamiltonian cycles in support of Kelly's conjecture.
متن کاملWhen Does Schwartz Conjecture Hold?
In 1990, Thomas Schwartz proposed the conjecture that every nonempty tournament has a unique minimal τ -retentive set (τ stands for tournament equilibrium set). A weak variant of Schwartz’s Conjecture was recently proposed by Felix Brandt. However, both conjectures were disproved very recently by two counterexamples. In this paper, we prove sufficient conditions for infinite classes of tourname...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discrete Mathematics
دوره 282 شماره
صفحات -
تاریخ انتشار 2004