tailieunhanh - Báo cáo toán học: "H-free graphs of large minimum degree"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: H-free graphs of large minimum degree. | H-free graphs of large minimum degree Noga Alon Benny Sudakov t Submitted Aug 16 2005 Accepted Feb 13 2006 Published Mar 7 2006 Mathematics Subject Classification 05C35 Abstract We prove the following extension of an old result of Andrasfai Erdos and Sós. For every fixed graph H with chromatic number r 1 3 and for every fixed e 0 there are n0 n0 H e and p p H 0 such that the following holds. Let G be an H-free graph on n n0 vertices with minimum degree at least 1 r_11 3 e n. Then one can delete at most n2_p edges to make G r-colorable. 1 Introduction Turan s classical Theorem 11 determines the maximum number of edges in a Kr 1 -free graph on n vertices. It easily implies that for r 2 if a Kr 1-free graph on n vertices has minimum degree at least 1 r n then it is r-colorable in fact it is a complete r-partite graph with equal color classes . The following stronger result has been proved by Andrasfai Erdos and Sos 2 . Theorem 2 If G is a Kr 1-free graph of order n with minimum degree d G 1 1 0 n then G is r-colorable. r 1 3 The following construction shows that this is tight. Let G be a graph whose vertex set is the disjoint union of r 3 sets U1 U2 . U5 and V1 V2 . Vr_2 in which Ui 3ỹ_yn for all i and Vj 3 1 n for all j. Each vertex of Vj is adjacent to all vertices but the other members of Vj and each vertex of Ui is adjacent to all vertices of U i 1 mod5 U i_1 mod5 Schools of Mathematics and Computer Science Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University Tel Aviv 69978 Israel and IAS Princeton NJ 08540 USA. Email no-gaa@. Research supported in part by a USA-Israeli BSF grant by NSF grant CCR-0324906 by a Wolfensohn fund and by the State of New Jersey. Department of Mathematics Princeton University Princeton NJ 08544 USA. E-mail bsu-dakov@. Research supported in part by NSF CAREER award DMS-0546523 NSF grant DMS-0355497 USA-Israeli BSF grant and by an Alfred P. Sloan fellowship. THE ELECTRONIC JOURNAL OF .