tailieunhanh - Báo cáo toán học: "Perfect matchings in -regular graphs"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Perfect matchings in -regular graphs. | Perfect matchings in e-regular graphs Noga Alon School of Mathematics Institute for Advanced Study Princeton NJ 08540 and Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University Tel Aviv Israel Email noga@. Vojtech Rodl Department of Mathematics and Computer Science Emory University Atlanta USA Email rodl@. Andrzej Rucinski Department of Discrete Mathematics Faculty of Mathematics and Computer Science Adam Mickiewicz University Poznan Poland Email rucinski@. Submitted December 10 1997 Accepted February 8 1998. Abstract A super d e -regular graph on 2n vertices is a bipartite graph on the classes of vertices V1 and V2 where V1 IV2I n in which the minimum degree and the maximum degree are between d e n and d e n and for every U c V1 W c V2 with IU I en W I en I j jj 1 e. We prove that for every 1 d 2e 0 and n n0 e the number of perfect matchings in any such graph is at least d 2e nn and at most d 2e nn . The proof relies on the validity of two well known conjectures for permanents the Minc conjecture proved by Bregman and the van der Waerden conjecture proved by Falikman and Egorichev. Research supported in part by a USA Israeli BSF grant by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University and by a State of New Jersey grant. Research supported by Polish-US NSF grant iNt-940671 and by NsF grant DMS-9704114. Research supported by Polish-US NSF grant INT-940671 and by KBN grant 2 P03A 023 09. 0Mathematics Subject Classification 1991 primary 05C50 05C70 secondary 05C80 1 e 1 THE ELECTRONIC .JOURNAL OF COmBINATORICS 5 1998 R13 2 An e-regular graph on 2n vertices is a bipartite graph on the classes of vertices V1 and V2 where V1 V2 n in which for every U c V1 W c V2 with UI en WI en e U W e Vi V -- --- --r -- U W IV1IIV2I where here e X Y denotes the number of edges between X and Y. The quantity jVnVd is called the density of the graph. Such a graph is a super d