Rocío Rojas-Monroy

Monochromatic paths and monochromatic sets of arcs in bipartite tournaments

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours and all of them are used. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices there is no monochromatic path between them and for every vertex v in V (D) \ N there is a monochromatic path from v to some vertex in N . We denote by A(u) the set of arcs of D that have u as the initial endpoint. In this paper we introduce the concept of semikernel modulo i by monochromatic paths of an m-coloured digraph. This concept allow us to find sufficient conditions for the existence of a kernel by monochromatic paths in an m-coloured digraph. In particular we deal with bipartite tournaments such that A(z) is monochromatic for each z ∈ V (D).

Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices of N there is no monochromatic path between them and for every vertex v / ∈ N there is a monochromatic path from v to N . We denote by A(u) the set of arcs of D that have u as the initial vertex. We prove that if D is an m-coloured 3-quasitransitive digraph such that for every vertex u of D, A(u) is monochromatic and D satisfies some colouring conditions over one subdigraph of D of order 3 and two subdigraphs of D of order 4, then D has a kernel by monochromatic paths.

Monochromatic cycles and monochromatic paths in arc-colored digraphs

We call the digraph D an m-colored digraph if the arcs of D are colored with m colors. A path (or a cycle) is called monochromatic if all of its arcs are colored alike. A cycle is called a quasi-monochromatic cycle if with at most one exception all of its arcs are colored alike. A subdigraph H in D is called rainbow if all its arcs have different colors. A set N ⊆ V (D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different vertices u, v ∈ N there is no monochromatic path between them and; (ii) for every vertex x ∈ V (D) − N there is a vertex y ∈ N such that there is an xymonochromatic path. The closure of D, denoted by C(D), is the m-colored multidigraph defined as follows: V (C(D)) = V (D), A(C(D)) = A(D) ∪ {(u, v) with color i | there exists a uv-monochromatic path colored i contained in D}. Notice that for any digraph D, C(C(D)) ∼= C(D) and D has a kernel by monochromatic paths if and only if C(D) has a kernel. Let D be a finite m−colored digraph. Suppose that there is a partition C = C1 ∪C2 of the set of colors of D such that every cycle in the subdigraph D[Ci] spanned by the arcs with colors in Ci is monochromatic. We show that if C(D) does not contain neither rainbow triangles nor rainbow P3 involving colors of both C1 and C2, then D has a kernel by monochromatic paths. This result is a wide extension of the original result by Sands, Sauer and Woodrow that asserts: Every 2−colored digraph has a kernel by monochromatic paths (since in this case there are no rainbow triangles in C(D)). 2010 Mathematics Subject Classification: 05C20.

On monochromatic paths and monochromatic 4-cycles in edge coloured bipartite tournaments

Abstract We call the digraph D an m -coloured digraph if the arcs of D are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A set N ⊆ V ( D ) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) For every pair of different vertices u , v ∈ N , there is no monochromatic directed path between them. (ii) For every vertex x ∈( V ( D )− N ), there is a vertex y ∈ N such that there is an xy -monochromatic directed path. In this paper it is proved that if D is an m -coloured bipartite tournament such that every directed cycle of length 4 is monochromatic, then D has a kernel by monochromatic paths.

Monochromatic paths and quasi-monochromatic cycles in edge-coloured bipartite tournaments

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A directed cycle is called quasi-monochromatic if with at most one exception all of its arcs are coloured alike. A set N ⊆ V (D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different vertices u, v ∈ N there is no monochromatic directed path between them and (ii) for every vertex x ∈ V (D)−N there is a vertex y ∈ N such that there is an xy-monochromatic directed path. In this paper it is proved that if D is an m-coloured bipartite tournament such that: every directed cycle of length 4 is quasi-monochromatic, every directed cycle of length 6 is monochromatic, and D has no induced particular 6-element bipartite tournament T̃6, then D has a kernel by monochromatic paths.

Monochromatic Paths and at Most 2-Coloured Arc Sets in Edge-Coloured Tournaments

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 C3 (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.

Kernels in pretransitive digraphs

Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively. A kernel N of D is an independent set of vertices such that for every w?V(D)-N there exists an arc from w to N. A digraph D is called right-pretransitive (resp. left-pretransitive) when (u,v)?A(D) and (v,w)?A(D) implies (u,w)?A(D) or (w,v)?A(D) (resp. (u,v)?A(D) and (v,w)?A(D) implies (u,w)?A(D) or (v,u)?A(D)). This concepts were introduced by P. Duchet in 1980. In this paper is proved the following result: Let D be a digraph. If D=D1?D2 where D1 is a right-pretransitive digraph, D2 is a left-pretransitive digraph and Di contains no infinite outward path for i?{1,2}, then D has a kernel.

γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs

Abstract We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. If D is an m-coloured digraph and a ∈ A(D), colour(a) will denote the colour has been used on a. A path (or a cycle) is called monochromatic if all of its arcs are coloured alike. A γ-cycle in D is a sequence of vertices, say γ = (u0, u1, . . . , un), such that ui ≠ uj if i ≠ j and for every i ∈ {0, 1, . . . , n} there is a uiui+1-monochromatic path in D and there is no ui+1ui-monochromatic path in D (the indices of the vertices will be taken mod n+1). A set N ⊆ V (D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different vertices u, v ∈ N there is no monochromatic path between them and; (ii) for every vertex x ∈ V (D) \ N there is a vertex y ∈ N such that there is an xy-monochromatic path. Let D be a finite m-coloured digraph. Suppose that {C1,C2} is a partition of C, the set of colours of D, and Di will be the spanning subdigraph of D such that A(Di) = {a ∈ A(D) | colour(a) ∈ Ci}. In this paper, we give some sufficient conditions for the existence of a kernel by monochromatic paths in a digraph with the structure mentioned above. In particular we obtain an extension of the original result by B. Sands, N. Sauer and R. Woodrow that asserts: Every 2-coloured digraph has a kernel by monochromatic paths. Also, we extend other results obtained before where it is proved that under some conditions an m-coloured digraph has no γ-cycles.

γ-Cycles In Arc-Colored Digraphs

Abstract We call a digraph D an m-colored digraph if the arcs of D are colored with m colors. A directed path (or a directed cycle) is called monochromatic if all of its arcs are colored alike. A subdigraph H in D is called rainbow if all of its arcs have different colors. A set N ⊆ V (D) is said to be a kernel by monochromatic paths of D if it satisfies the two following conditions: for every pair of different vertices u, v ∈ N there is no monochromatic path in D between them, and for every vertex x ∈ V (D) − N there is a vertex y ∈ N such that there is an xy-monochromatic path in D. A γ-cycle in D is a sequence of different vertices γ = (u0, u1, . . . , un, u0) such that for every i ∈ {0, 1, . . . , n}: there is a uiui+1-monochromatic path, and there is no ui+1ui-monochromatic path. The addition over the indices of the vertices of γ is taken modulo (n + 1). If D is an m-colored digraph, then the closure of D, denoted by ℭ(D), is the m-colored multidigraph defined as follows: V (ℭ (D)) = V (D), A(ℭ (D)) = A(D) ∪ {(u, v) with color i | there exists a uv-monochromatic path colored i contained in D}. In this work, we prove the following result. Let D be a finite m-colored digraph which satisfies that there is a partition C = C1 ∪ C2 of the set of colors of D such that: D[Ĉi] (the subdigraph spanned by the arcs with colors in Ci) contains no γ-cycles for i ∈ {1, 2}; If ℭ (D) contains a rainbow C3 = (x0, z, w, x0) involving colors of C1 and C2, then (x0, w) ∈ A(ℭ (D)) or (z, x0) ∈ A(ℭ (D)); If ℭ (D) contains a rainbow P3 = (u, z, w, x0) involving colors of C1 and C2, then at least one of the following pairs of vertices is an arc in ℭ (D): (u, w), (w, u), (x0, u), (u, x0), (x0, w), (z, u), (z, x0). Then D has a kernel by monochromatic paths. This theorem can be applied to all those digraphs that contain no γ-cycles. Generalizations of many previous results are obtained as a direct consequence of this theorem.

H-paths in 2-colored tournaments

Let G and H be two digraphs. G is H−colored when there exists a function f : A(G) −→ V (H). A walk C = (v1, v2, . . . , vn) in G is an H−walk if the list (f(v1, v2), ..., f(vn−1, vn)) is a walk in H. In addition, let N ⊆ V (G) N is a kernel by H−walks if the following conditions are held: 1) If x ∈ V (G) − N, then there exists an x, y − H−walk for some y ∈ N . 2) Let x, y ∈ N, with x 6= y, then there is no x, y −H−walk in G. When every induced subdigraph of D has a kernel by H−walks we say that D is kernel perfect by H−walks. In this work we introduce the following result: Let H be a digraph such that |V (H)| ≤ 2, and suppose that H is not isomorphic to the tournament of 2 vertices. For every H−colored tournament T such that every C3 (a cycle of length 3) in T contains an H−walk of length of at least 2, then T is kernel perfect by H−walks. This result generalizes the one obtained by Sands, Sauer and Woodrow: Every 2−colored tournament has a vertex x such that for every other vertex u in the tournament there exists a ux−monochromatic path where all of its arcs are colored alike. Mathematics Subject Classification: 05C20 186 Alejandro Contreras-Balbuena and Roćıo Rojas-Monroy