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The maximum number of perfect matchings in graphs with a given degree sequence
 

Summary: The maximum number of perfect matchings in graphs
with a given degree sequence
Noga Alon
Shmuel Friedland
17 March, 2008
Abstract
We show that the number of perfect matching in a simple graph G with an
even number of vertices and degree sequence d1, d2, . . . , dn is at most
n
i=1(di!)
1
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 V , let deg v denote
its degree. Assume that |V | is even, and let perfmat G denote the number of perfect
matchings in G. The main result of this short note is:

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics