a b s t r a c tLet D be a digraph. V (D) and A(D) will denote the vertex and arc sets of D respectively. A kernel K of a digraph D is an independent set of vertices of D such that for every vertex w in V (D)-K there exists an arc from w to a vertex in K . Let H be a digraph possibly with loops and D a digraph without loops whose arcs are colored with the vertices of H (D is said to be an H-colored digraph). A directed path W in D is said to be an H-path if and only if the consecutive colors encountered on W form a directed walk in H. A generalization of the concept of kernel is the concept of H-kernel, where an H-kernel N of an H-colored digraph D is a set of vertices of D such that for every pair of different vertices in N there is no H-path between them, and for every vertex u in V (D)-N there exists an H-path in D from u to N.A classical result in kernel theory establishes that if D is a digraph without cycles of odd length, then D has a kernel; this result is known as Richardson's theorem and in this paper we will show an extension of this theorem which is given by the main result.Let D be an H-colored digraph. For an arc (z 1 , z 2 ) of D we will denote its color by c(z 1 , z 2 ). We introduce the concept of obstruction in an H-colored digraph as follows. Let W = (v 0 , v 1 , . . . , v n−1 , v 0 ) be a closed directed walk in D. We will say that there is an obstructionis not an arc of A(H) (indices modulo n). The main result establishes that if D is an H-colored digraph such that the number of obstructions in every closed directed trail of D is even, then D has an H-kernel. Previous interesting results are generalized, as for example Sands, Sauer and Woodrow's theorem.

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Let H be a digraph possibly with loops and D a finite digraph without loops whose arcs are coloured with the vertices of H (D is an H -coloured digraph). The sets V(D) and A(D) will denote the sets of vertices and arcs of D respectively. A directed path W in D is an H -path if and only if the consecutive colors encountered on W 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 . Let D be an m-coloured digraph. The color-class digraph of D, denoted by C C (D), is the digraph such that: the vertices of the color-class digraph are the colors represented in the arcs of D, and (i, j) ∈ A(C C (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. Let W = (v 0 , . . . , v n ) be a directed walk in C C (D), with D an H -coloured digraph, and e i = (v i , v i+1 ) for each i ∈ {0, . . . , n − 1}. Let I = {i 1 , . . . , i k } a subset of {0, . . . , n − 1} such that for 0 ≤ s ≤ n − 1, e s ∈ A(H c ) if and only if s ∈ I (where H c is the complement of H ), then we will say that k is the H c -length of W . Since V(C C (D)) ⊆ V(H ), the main question is: What structural properties of C C (D), with respect to H , imply that D has an H -kernel? In this paper we will prove the following: If C C (D) does not have directed cycles of odd H c -length, then D has an H -kernel. Finally we will prove Richardson's theorem as a direct consequence of the previous result.

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