tailieunhanh - Báo cáo toán học: "omputation of the vertex Folkman numbers F (2, 2, 2, 4; 6) and F (2, 3, 4; 6)"

CTuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: omputation of the vertex Folkman numbers F (2, 2, 2, 4; 6) and F (2, 3, 4; 6). | Computation of the vertex Folkman numbers F 2 2 2 4 6 and F 2 3 4 6 Evgeni Nedialkov Institute of Mathematics and Informatics Bulgarian Academy of Sciences MOI 8 Acad. G. Bonchev Str. 1113 Sofia BULGARIA nedialkov@ Nedyalko Nenov Faculty of Mathematics and Informatics Sofia University 5 James Baurchier str. Sofia BULGARIA nenov@ Submitted August 30 2001 Accepted February 26 2002. MR Subject Classifications 05C55 Abstract In this note we show that the exact value of the vertex Folkman numbers F 2 2 2 4 6 and F 2 3 4 6 is 14. 1 Notations We consider only finite non-oriented graphs without loops and multiple edges. The vertex set and the edge set of a graph G will be denoted by V G and E G respectively. We call p-clique of G any set of p vertices each two of which are adjacent. The largest natural number p such that the graph G contains a p-clique is denoted by cl G the clique number of G . A set of vertices of a graph G is said to be independent if every two of them are not adjacent. The cardinality of any largest independent set of vertices in G is denoted by a G the independence number of G . If W c V G then G W is the subgraph of G induced by W and G W is the subgraph induced by V G W. We shall use also the following notation G - the complement of graph G Kn - complete graph of n vertices Cn - simple cycle of n vertices N v - the set of all vertices adjacent to v THE ELECTRONIC JOURNAL OF COMBINATORICS 9 2002 R9 1 ỵ G - the chromatic number of G Kn Cm m n - the graph obtained from Kn by deleting all edges of some cycle Cm. 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 EG u E where E x y x 2 V G1 y 2 V G2 . 2 Vertex Folkman numbers. Definition 1. Let G be a graph and let a1 . ar be positive integers r 2. An r-coloring V G Vi u. u Vr Vi n Vj 0 i j of the vertices of G is said to be a1 . . . ar -free if for all i 2 1 . r the graph G does not contain a .