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In this paper, we give sufficient conditions for the existence of kernels by monochromatic directed paths (m.d.p.) in digraphs with quasi-transitive colorings. Let D be an m-colored digraph. We prove that if every chromatic class of D is quasi-transitive, every cycle is quasi-transitive in the rim and D does not contain polychromatic triangles, then D has a… (More)

The mean value of the matching polynomial is computed in the family of all labeled graphs with n vertices. We introduce the dominating polynomial of a graph whose coefficients enumerate the dominating sets for a graph and study some properties of the polynomial. The mean value of this polynomial is determined in a certain special family of bipartite… (More)

We study k-colored kernels in m-colored digraphs. An m-colored digraph D has k-colored kernel if there exists a subset K of its vertices such that (i) from every vertex v / ∈ K there exists an at most k-colored directed path from v to a vertex of K and (ii) for every u, v ∈ K there does not exist an at most k-colored directed path between them. In this… (More)