H-Kernels in Unions of H-Colored Quasi-Transitive Digraphs

Let be a digraph (possibly with loops) and a digraph without loops whose arcs are colored with the vertices of ( is said to be an -colored digraph). For an arc ( ) of , its color is denoted by ). A directed path = ( , . . ., ) in an -colored digraph will be called -path if and only if ( , ), . . ., , )) is a directed walk in . In , we will say that there is an obstruction on if ( , ), )) ∉ ) (if = we will take indices modulo ). A subset of ( ) is said to be an -kernel in if for every pair of di erent vertices in there is no -path between them, and for every vertex in ( ) \ there exists an -path in from to . Let be an arc-colored digraph. The color-class digraph of , C ), is the digraph such that (C )) = { ) : ∈ )} and ( ) ∈ (C )) if and only if there exist two arcs, namely ( ) and ( ) in , such that ) = and ) = . The main result establishes that if = ∪ is an -colored digraph which is a union of asymmetric quasi-transitive digraphs and { , . . ., } is a partition of (C )) with a property such that 1. is a quasi-transitive -class for every in {1, . . ., }, 2. either [{ ∈ ) : ) ∈ }] is a subdigraph of or it is a sudigraph of for every in {1, . . ., }, 3. has no infinite outward path for every in {1, 2}, 4. every cycle of length three in has at most two obstructions, then has an -kernel. Some results with respect to the existence of kernels by monochromatic paths in finite digraphs will be deduced from the main result.