Cycles and transitivity by monochromatic paths in arc-coloured digraphs

A digraph D is an m-coloured digraph if its arcs are coloured with m colours. If D is an m-coloured digraph and a∈A(D), then colour(a) will denote the colour has been used on a. A path (or a cycle) is monochromatic if all of its arcs are coloured alike. A set N⊆V(D) is 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. The closure of D, denoted by C(D), is the m-coloured multidigraph defined as follows: V(C(D))=V(D), A(C(D))=A(D)∪{(u,v) with colour i| there is an uv-path coloured i contained in D}. A subdigraph H in D is rainbow if all of its arcs have different colours. A digraph D is transitive by monochromatic paths if the existence of an xy-monochromatic path and an yz-monochromatic path in D imply that there is an xz-monochromatic path in D. We will denote by P3⃗ the path of length 3 and by C3⃗ the cycle of length 3. Let D be a finite m-coloured digraph. Suppose that C is the set of colours used in A(D), and ζ={C1,C2,…,Ck} (k≥2) is a partition of C, such that for every i∈{1,2,…,k} happens that Hi=D[{a∈A(D)∣colour(a)∈Ci}] is transitive by monochromatic paths. Let {ζ1,ζ2} be a partition of ζ, and Di will be the spanning subdigraph of D such that A(Di)={a∈A(D)∣colour(a)∈CjforsomeCj∈ζi}. 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 extensions of the following two original results: The result by Sands et al. (1982) that asserts: Every 2-coloured digraph has a kernel by monochromatic paths, and the result by Galeana-Sánchez et al. (2011) that asserts: If D is a finite m-coloured digraph that admits a partition {C1,C2} of the set of colours of D such that for each i∈{1,2} every cycle in the subdigraph D[Ci] spanned by the arcs with colours in Ci is monochromatic, C(D) does not contain neither rainbow triangles nor rainbow P3⃗ (path of length 3) involving colours of both C1 and C2; then D has a kernel by monochromatic paths