H-Kernels in Infinite Digraphs

Let H be a digraph possibly with loops and D a digraph (possibly infinite) without loops whose arcs are coloured with the vertices of H (D is an H-coloured digraph). V(D) and A(D) will denote the sets of vertices and arcs of D respectively. A directed walk or a directed pathW in D is an H-walk or an H-path if and only if the consecutive colors encountered onW form a directed walk in H. A set N ⊆ V(D) is an H-kernel if for every pair of different vertices in N there is no H-path between them, and for every vertex u ∈ V(D)\N there exists an H-path in D from u to N. Linek and Sands introduced the concept of H-walk and this concept was later used by several authors. In particular, Galeana-Sánchez and Delgado-Escalante used the concept of H-walk in order to introduce the concept of H-kernel, which generalizes the concepts of kernel and kernel by monochromatic paths. Let D be an arc-coloured digraph. In 2009 Galeana-Sánchez introduced the concept of color-class digraph of D, denoted by CC(D), as follows: the vertices of the color-class digraph are the colors represented in the arcs of D, and (i, j) ∈ A(CC(D)) if and only if there exist two arcs namely (u,v) and (v,w) in D such that (u,v) has color i and (v,w) has color j. Since V(CC(D)) ⊆ V(H), the main question is: What structural properties of CC(D), with respect toH, imply thatD has anH-kernel? Suppose thatD has no infinite outward H-path. In this paper we prove that if CC(D) ⊆ H, then D has an H-kernel. We also prove that if there exists a partition (V1,V2) of V(CC(D)) such that: (1) CC(D)[Vi] ⊆ H[Vi] for each i ∈ {1,2}, (2) if (u,v) ∈ A(CC(D)) for some u ∈ Vi and for some v ∈ Vj, with i ̸= j and i, j ∈ {1,2}, then (u,v) / ∈ A(H), and (3) D has no Vi-colored infinite outward H-path for each i ∈ {1,2}. Then D has an H-kernel. Several previous results are generalized. H. Galeana-Sánchez, Rocı́o Sánchez-López Instituto de Matemáticas, UNAM Ciudad Universitaria, Circuito Exterior 04510 México, D.F., México E-mail: [email protected] R. Sánchez-López E-mail: [email protected] 2 Hortensia Galeana-Sánchez, Rocı́o Sánchez-López