Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: New upper bound for a class of vertex Folkman numbers. | New upper bound for a class of vertex Folkman numbers N. Kolev Department of Algebra Faculty of Mathematics and Informatics St. Kl. Ohridski University of Sofia 5 J. Bourchier blvd 1164 Sofia BULGARIA N. Nenov Department of Algebra Faculty of Mathematics and Informatics St. Kl. Ohridski University of Sofia 5 J. Bourchier blvd 1164 Sofia BULGARIA nenov@ Submitted Jun 9 2005 Accepted Feb 7 2006 Published Feb 15 2006 Mathematics Subject Classification 05C55 Abstract Let a1 . ar be positive integers m r r 1 aj 1 1 and p max ai . 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 aj-clique of color i for some i 1 . r. The vertex Folkman numbers F a1 . ar m 1 min u G G a1 . .ar and Km-1 G are considered. We prove that F a1 . ar m 1 m 3p p 3. This inequality improves the bound for these numbers obtained by Luczak Rucinski and Urbanski 2001 . 1 Introduction We consider only finite non-oriented graphs without loops and multiple edges. We call a p-clique of the graph G a set of p vertices each two of which are adjacent. The largest positive integer p such that the graph G contains a p-clique is denoted by cl G . In this paper we shall also use the following notations V G - vertex set of the graph G E G - edge set of the graph G THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R14 1 G - the complement of G G V V c V G - the subgraph of G induced by V G V - the subgraph induced by the set V G V Ng v v 2 V G - the set of all vertices of G adjacent to v Kn - the complete graph on n vertices Cn - simple cycle on n vertices Pn - path on n vertices x G - the chromatic number of G tx - the least positive integer greater or equal to x. Let G1 and G2 be two graphs without common vertices. We denote by G1 G2 the graph G for which V G V G1 u V G2 and E G E G1 u E g2 u E Where E x y I x 2 V G1 y 2 V G2 . Definition Let a1 . ar be positive integers. We say that the r-coloring V G V1 u. u Vr Vi n Vj 1 j
