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

Casas-Bautista, Enrique  ; Galeana-Sánchez, Hortensia  ; Rojas-Monroy, Rocío 

Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 3, pp. 493-507

Résumé

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, . . ., u_n), such that u_i ≠ u_j if i ≠ j and for every i ∈ 0, 1, . . ., n there is a u_iu_i+1-monochromatic path in D and there is no u_i+1u_i-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 D_i will be the spanning subdigraph of D such that A(D_i) = a ∈ A(D) | colour(a) ∈ C_i. 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.

Détail

Keywords: digraph, kernel, kernel by monochromatic paths, γ-cycle