Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Steiner triple systems and existentially closed graphs. | Steiner triple systems and existentially closed graphs . Forbes . Grannell and . Griggs Department of Pure Mathematics The Open University Walton Hall Milton Keynes MK7 6AA united kingdom tonyforbes@ Submitted Dec 20 2004 Accepted Apr 7 2005 Published Aug 30 2005 Mathematics Subject Classifications 05C99 05B07 Abstract We investigate the conditions under which a Steiner triple system can have a 2- or 3-existentially closed block intersection graph. 1 Introduction A graph G V E where V is the set of vertices and E is the set of edges is said to be n-existentially closed or . if for every n-element subset S of V and for every subset T of S there exists a vertex x 2 S which is adjacent to every vertex in T and is not adjacent to any vertex in S n T. These graphs were first studied by Caccetta Erdos and Vijayan 6 although Erdos and Rényi 8 had previously proved the interesting result that for any fixed value of n almost all graphs are . But relatively few specific examples of . graphs are known for n 2. A strongly regular graph SRG v k X p is a regular graph of degree k on v vertices with the property that every pair of adjacent vertices has X common neighbours and every pair of non-adjacent vertices have p common neighbours. An important class of strongly regular graphs are the Paley graphs. The Paley graph of order q where q 1 mod 4 is a prime power is the graph with vertex set GF q the Galois field of order q and the edge set is the set of pairs x y where x y is a square. It is an SRG q q 1 2 q 5 4 q 1 4 . Ananchuen and Caccetta 1 proved that all Paley graphs with at least 29 vertices are . see also 3 and 5 . In 2 Baker Bonato and Brown constructed . strongly regular graphs SRG q2 q2 1 2 q2 5 4 q2 1 4 THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R42 1 using affine planes of order q 7. In 4 Bonato Holzmann and Kharagani studied . strongly regular graphs obtained .
