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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Induced complete h-partite graphs in dense clique-less graphs. | Induced complete h-partite graphs in dense clique-less graphs Eldar Fischer Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel-Aviv University Tel-Aviv Israel. eldar@math.tau.ac.il Submitted June 3 1999 Accepted September 16 1999. Abstract It is proven that for every fixed h a and b a graph with n vertices and minimum degree at least h- 1 n which contains no copy of Kb the complete graph with b vertices contains at least 1 o 1 ha vertex disjoint induced copies of the complete h-partite graph with a vertices in each color class. Mathematics subject classification numbers 05C70 05C55. All graphs considered here are finite undirected and have neither loops nor parallel edges. The notation here follows the convention of 4 except where stated otherwise. Many asymptotic embedding results have been proven by the Regularity Lemma of Szemeredi 6 over the years. See 5 for a survey. One of the results in this area that of Alon and Yuster 2 has particular relevance to the following. Theorem 1 2 For every natural a h and every e 0 there exists N N h a e such that if G is a graph with n N vertices and minimum degree at least n then G contains at least 1 e ha vertex disjoint copies of K h a the complete h-partite graph with a vertices in each color class. Research supported by the Fritz Brann Doctoral Fellowship in Engineering and Exact Sciences. 1 THE ELECTRONIC .JOURNAL OF COmBINATORICS 6 1999 R43 2 The following result proven below is an analogue of Theorem 1 for the context of induced subgraphs. Theorem 2 For every h a b and e 0 there exists N N h a b e such that if G is a graph with n N vertices minimum degree at least h-1 n and no copy of Kb the complete graph with b vertices then G contains at least 1 e ha vertex disjoint induced copies of K h a the complete h-partite graph with a vertices in each color class. In 1 a variant of the Regularity Lemma which is suitable for dealing with induced subgraphs in very general circumstances is .
