Counting 1-Factors in Regular Bipartite Graphs

We show that any k-regular bipartite graph with 2n vertices has at least ((k−1)k−1 kk−2 )n perfect matchings (1-factors). Equivalently, this is a lower bound on the permanent of any nonnegative integer n × n matrix with each row and column sum equal to k. For any k, the base (k−1)k−1 kk−2 is largest possible.

• Type: Journal
• Year: 1998
• Where: JCT
• Authors: Alexander Schrijver
• Keywords: JCT 1998, K-regular Bipartite Graph, Lower Bound, Nonnegative Integer