In this paper, we introduce a generalization of a kernel for vertex-colored digraphs. Let D be a digraph, we say that a function \(c:V(D)\rightarrow \{0,1,\ldots \}\) is a c-coloring of D. An up-color kernel of D, K, is a set of vertices that holds: i)for every vertex v in \(V(D)\setminus K\) , there exists a vertex w in K, which is adjacent to v and \(c(v)<c(w)\) , ii)K does not contain any vertex having color 0, and iii)every pair of vertices in K is not adjacent, that is, K is an independent set. We give sufficient and necessary conditions for some families of digraphs (directed paths, directed cycles, forests, wheels, digraphs which have a unique cycle, cycles with chords), as well as for certain products of digraphs (the Cartesian and strong products, the Zykov sum, the generalization of crown and the line digraph), to have an up-color kernel.
