8 research outputs found
Chromatic number of graphs and edge Folkman numbers
In the paper we give a lower bound for the number of vertices of a given
graph using its chromatic number. We find the graphs for which this bound is
exact. The results are applied in the theory of Foklman numbers.Comment: 9 pages, 1 figur
On a Class of Vertex Folkman Numbers
Let a1 , . . . , ar, be positive integers, i=1 ... r, m = β(ai β 1) + 1 and
p = max{a1 , . . . , ar }. For a graph G the symbol G β (a1 , . . . , ar ) means
that in every r-coloring of the vertices of G there exists a monochromatic
ai -clique of color i for some i β {1, . . . , r}. In this paper we consider the
vertex Folkman numbers
F (a1 , . . . , ar ; m β 1) = min |V (G)| : G β (a1 , . . . , ar ) and Kmβ1 β G}
We prove that F (a1 , . . . , ar ; m β 1) = m + 6, if p = 3 and m β§ 6 (Theorem
3) and F (a1 , . . . , ar ; m β 1) = m + 7, if p = 4 and m β§ 6 (Theorem 4)
On the Nonexistence of Some Generalized Folkman Numbers
For an undirected simple graph , we write if and only if for every red-blue coloring of its vertices there exists a red or a blue . The generalized vertex Folkman number is defined as the smallest integer for which there exists an -free graph of order such that . The generalized edge Folkman numbers are defined similarly, when colorings of the edges are considered. We show that and are well defined for . We prove the nonexistence of for some , in particular for , where is the book graph of triangular pages, and for . We pose three problems on generalized Folkman numbers, including the existence question of edge Folkman numbers , and . Our results lead to some general inequalities involving two-color and multicolor Folkman numbers