20 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 the 3-Colouring Vertex Folkman Number F(2,2,4)
In this note we prove that F (2, 2, 4) = 13
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)