Digital Object Identifier (DOI) 10.1007/s00373-005-0618-z
Graphs and Combinatorics (2005) 21:307–317
Graphs and
Combinatorics
© Springer-Verlag 2005
Monochromatic Paths and at Most 2-Coloured Arc Sets
in Edge-Coloured Tournaments
Hortensia Galeana-S ´ anchez
1
and Roc´ ıo Rojas-Monroy
2
1
Instituto de Matem´ aticas, UNAM, Ciudad Universitaria, Circuito Exterior, 04510 M´ exico,
D.F., M´ exico. e-mail: hgaleana@matem.unam.mx
2
Facultad de Ciencias, Universidad Aut´ onoma del Estado de M´ exico, Instituto Literario
No. 100, Centro, 50000, Toluca, Edo. de M´ exico, M´ exico
Abstract. We call the tournament T an m-coloured tournament if the arcs of T are coloured
with m-colours. If v is a vertex of an m-coloured tournament T , we denote by ξ(v) the set of
colours assigned to the arcs with v as an endpoint.
In this paper is proved that if T is an m-coloured tournament with |ξ(v)|≤ 2 for each
vertex v of T , and T satisfies at least one of the two following properties (1) m = 3 or (2)
m = 3 and T contains no C
3
(the directed cycle of length 3 whose arcs are coloured with
three distinct colours). Then there is a vertex v of T such that for every other vertex x of T ,
there is a monochromatic directed path from x to v.
Key words. Kernel, Kernel-perfect digraph, Kernel by monochromatic paths, Tournament,
m-Coloured tournament
1. Introduction
For general concepts we refer the reader to [1]. Let D be a digraph; V (D) and
A(D) will denote the sets of vertices and arcs of D, respectively. An arc (u
1
,u
2
) ∈
A(D) is called asymmetrical (resp. symmetrical) if (u
2
,u
1
)/ ∈ A(D) (resp. (u
2
,u
1
) ∈
A(D)). The asymmetrical part of D (resp. symmetrical part of D) which is de-
noted Asym(D) (resp. Sym(D)), is the spanning subdigraph of D whose arcs are
the asymmetrical (resp. symmetrical) arcs of D; D is called an asymmetrical digraph
if Asym(D) = D. We recall that a subdigraph D
1
of D is a spanning subdigraph if
V (D
1
) = V (D).
If S is a nonempty set of V (D) then the subdigraph D[S ] induced by S is the
digraph with vertex set S and those arcs of D which join vertices of S . An arc (u
1
,u
2
)
of D will be called an S
1
S
2
-arc whenever u
1
∈ S
1
and u
2
∈ S
2
.
A set I ⊆ V (D) is independent if A(D[I ]) =∅. A kernel N of D is an inde-
pendent set of vertices such that for each z ∈ (V (D) - N) there exists a zN -arc in
D. A digraph D is called kernel-perfect digraph or KP -digraph when every induced
subdigraph of D has a kernel.
Mathematics Subject Classification (2000): 05C20