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In this paper, we consider the following problem due to Erd˝ os: for each m ∈ N, is there a (least) positive integer f (m) so that every finite m-colored tournament contains an absorbent set S by monochromatic directed paths of f (m) vertices? In particular, is f (3) = 3? We prove several bounds for absorbent sets of m-colored tournaments under certain… (More)

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)

Given a finite set P ⊆ R d , called a pattern, t P (n) denotes the maximum number of translated copies of P determined by n points in R d. We give the exact value of t P (n) when P is a rational simplex, that is, the points of P are rationally affinely independent. In this case, we prove that t P (n) = n − m r (n), where r is the rational affine dimension… (More)

In this paper, we construct infinite families of tight regular tournaments. In particular, we prove that two classes of regular tournaments, tame molds and ample tournaments are tight. We exhibit an infinite family of 3-dichromatic tight tournaments. With this family we positively answer to one case of a conjecture posed by V. Neumann-Lara. Finally, we show… (More)