3 M ar 2 00 9 On the Vertex Folkman Numbers
نویسنده
چکیده
For a graph G the symbol G v → (a1, . . . , ar) means that in every r-coloring of the vertices of G for some i ∈ {1, . . . , r} there exists a monochromatic ai-clique of color i. The vertex Folkman numbers Fv(a1, . . . , ar; q) = min{|V(G)| : G v → (a1, . . . , ar) and Kq " G} are considered. In this paper we shall compute the Folkman numbers Fv(2, . . . , 2 } {{ } r ; r− k + 1) when k ≤ 12 and r is sufficiently large. We prove also new bounds for some vertex and edge Folkman numbers. 2000 Mathematics Subject Classification: 05C55
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New Recurrent Inequality on a Class of Vertex Folkman Numbers
Let G be a graph and V (G) be the vertex set of G. Let a1 ,. . . , ar be positive integers, m = ∑ r i=1 (ai−1)+1 and p = max{a1, . . . , ar}. The symbol G → {a1, . . . , ar} denotes that in every r-coloring of V (G) there exists a monochromatic ai-clique of color i for some i = 1, . . . , r. The vertex Folkman numbers F (a1, . . . , ar ; m − 1) = min{|V (G)| : G → (a1 . . . ar) and Km−1 6⊆ G} a...
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