Richardson’s Theorem in H-coloured Digraphs

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 $$N\subseteq \hbox {V}(D)$$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 $$u\in \hbox {V}(D){\setminus }N$$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 $${\mathscr {C}}_C(D$$CC(D), is the digraph such that: the vertices of the color-class digraph are the colors represented in the arcs of D, and $$(i,j) \in A({\mathscr {C}}_C(D$$(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. Let $$W=(v_0, \ldots , v_n$$W=(v0,…,vn) be a directed walk in $${\mathscr {C}}_C(D)$$CC(D), with D an H-coloured digraph, and $$e_i = (v_{i},v_{i+1})$$ei=(vi,vi+1) for each $$i \in \{0, \ldots ,n-1\}$$i∈{0,…,n-1}. Let $$I = \{i_1, \ldots , i_k\}$$I={i1,…,ik} a subset of $$\{0, \ldots , n-1\}$${0,…,n-1} such that for 0 $$\le s \le n-1$$≤s≤n-1, $$e_s \in \hbox { A}(H^c)$$es∈A(Hc) if and only if $$s \in I$$s∈I (where $$H^c$$Hc is the complement of H), then we will say that k is the $$H^c$$Hc-length of W. Since V($${\mathscr {C}}_C(D)) \subseteq \hbox {V}(H)$$CC(D))⊆V(H), the main question is: What structural properties of $${\mathscr {C}}_C(D)$$CC(D), with respect to H, imply that D has an H-kernel? In this paper we will prove the following: If $${\mathscr {C}}_C(D)$$CC(D) does not have directed cycles of odd $$H^c$$Hc-length, then D has an H-kernel. Finally we will prove Richardson’s theorem as a direct consequence of the previous result.