tailieunhanh - Báo cáo toán học: "The maximum number of perfect matchings in graphs with a given degree sequence"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The maximum number of perfect matchings in graphs with a given degree sequence. | The maximum number of perfect matchings in graphs with a given degree sequence Noga Alon Shmuel Friedlandyz Submitted Mar 19 2008 Accepted Apr 13 2008 Published Apr 24 2008 Abstract We show that the number of perfect matchings in a simple graph G with an even 1 number of vertices and degree sequence d1 d2 . dn is at most Idn i di 2di This bound is sharp if and only if G is a union of complete balanced bipartite graphs. 2000 Mathematics Subject Classification 05A15 05C70. Keywords and phrases Perfect matchings permanents. 1 Introduction Let G V E be an undirected simple graph. For a vertex v 2 V let deg v denote its degree. Assume that VI is even and let perfmat G denote the number of perfect matchings in G. The main result of this short note is Theorem 1 perfmat G ỊỊ deg v 2dv v2V where 01 0. If G has no isolated vertices then equality holds if and only if G is a disjoint union of complete balanced bipartite graphs. For bipartite graphs the above inequality follows from the Bregman-Minc Inequality for permanents of 0 1 matrices mentioned below. The inequality was known to Kahn and Lovasz . 2 7 but their proof was never published and it was recently stated and proved independently by the second author in 3 . Here we show that it is a simple consequence of the Bregman-Minc Inequality. School of Mathematics Tel Aviv University Ramat Aviv Tel Aviv 69978 Israel and IAS Princeton NJ 08540 USA e-mail nogaa@. Research supported in part by the Israel Science Foundation and by a USA-Israeli BSF grant. yDepartment of Mathematics Statistics and Computer Science University of Illinois at Chicago Chicago Illinois 60607-7045 USA e-mail friedlan@ zVisiting Professor Fall 2007 - Winter 2008 Berlin Mathematical School Berlin Germany THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N13 1 2 The proof Let A be an n X n 0 1 matrix . A aj j i 2 0 1gnxn. Denote Vi Pn 1 Oij i 1 . n. The celebrated Bregman-Minc inequality conjectured by Mine 4 and .