New Recurrent Inequality on a Class of Vertex Folkman Numbers
نویسندگان
چکیده
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} are considered. In this paper we improve the known upper bounds on the numbers F (2, 2, p;p + 1) and F (3, p; p + 1).
منابع مشابه
New Upper Bound for a Class of Vertex Folkman Numbers
Let a1, . . . , ar be positive integers, m = ∑r i=1(ai−1)+1 and p = max{a1, . . . , ar}. For a graph G the symbol G → {a1, . . . , ar} denotes that in every r-coloring of the vertices of 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} are considered. We prove that...
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For graph G and integers a1 ≥ · · · ≥ ar ≥ 2, we write G → (a1, · · · , ar) if and only if for every r-coloring of the vertex set V (G) there exists a monochromatic Kai in G for some color i ∈ {1, · · · , r}. The vertex Folkman number Fv(a1, · · · , ar; s) is defined as the smallest integer n for which there exists a Ks-free graph G of order n such that G→ (a1, · · · , ar). It is well known tha...
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