An upper bound for the number of perfect matchings in graphs

Electronic Journal of Linear Algebra, 2012

This survey paper deals with upper and lower bounds on the number of k-matchings in regular graphs on N vertices. For the upper bounds we recall the upper matching conjecture which is known to hold for perfect matchings. For the lower bounds we first survey the known results for bipartite graphs, and their continuous versions as the van der Waerden and Tverberg permanent conjectures and its variants. We then discuss non-bipartite graphs. Little is known beyond the recent proof of the Lovász-Plummer conjecture on the exponential growth of perfect matchings in cubic bridgeless graphs. We discuss the problem of the minimum of haffnians on the convex set of matrices, whose extreme points are the adjacency matrices of subgraphs of the complete graph corresponding to perfect matchings. We also consider infinite regular graphs. The analog of k-matching is the p-monomer entropy, where p ∈ [0, 1] is the density of the number of matchings.

Discrete Mathematics, 1989

Computing Research Repository - CORR, 2006

We show that the number of k-matching in a given undirected graph G is equal to the number of perfect matching of the corresponding graph Gk on an even number of vertices divided by a suitable factor. If G is bi- partite then one can construct a bipartite Gk. For bipartite graphs this result implies that the number of k-matching has a polynomial-time ap- proximation algorithm. The above results are extended to permanents and hafnians of corresponding matrices.

2018

Inspired by some recent revisitations of the Cantor-Bernstein theorem, in particular its formalizations in ZF carried out via the proof assistant AProS by W. Sieg and P. Walsh, we are carrying out the proof of a related graph-theoretical proposition. Our development is assisted by the proof checker ÆtnaNova, and our proof pattern is drawn from Halmos’s classic ‘Naive set theory’. This case-study illustrates the flexibility of a proof environment rooted in Set Theory, which can be bent with equal ease toward declarative and procedural styles of proof.

2011

Number of k -matchings in bipartite graphs and graphs as permanents and haffnians Upper bounds on permanents and haffnians: results and conjectures. Lower bounds on permanents and haffnians: results and conjectures. Shmuel Friedland Univ. Illinois at Chicago () Some open problems in matchings in graphs Stability, hyperbolicity, and zero localization of / 29 Matchings G = (V , E ) undirected graph with vertices V , edges E . matching in G: M ⊆ E no two edges in M share a common endpoint. e = (u, v) ∈ M is dimer v not covered by M is monomer. M called monomer-dimer cover of G. M is perfect matching ⇐⇒ no monomers. M is k-matching ⇐⇒ #M = k. Shmuel Friedland Univ. Illinois at Chicago () Some open problems in matchings in graphs Stability, hyperbolicity, and zero localization of / 29

The Electronic Journal of Combinatorics

We give upper bounds on weighted perfect matchings in Pfaffian graphs. These upper bounds are better than Bregman's upper bounds on the number of perfect matchings. We show that some of our upper bounds are sharp for 3 and 4-regular Pfaffian graphs. We apply our results to fullerene graphs.

SIAM Journal on Discrete Mathematics, 2017

Let f (n) be the smallest number such that every collection of n matchings, each of size at least f (n), in a bipartite graph, has a full rainbow matching. Generalizing famous conjectures of Ryser, Brualdi and Stein, Aharoni and Berger [1] conjectured that f (n) = n + 1 for every n > 1. Clemens and Ehrenmüller [4] proved that f (n) ≤ 3 2 n + o(n). We show that the o(n) term can be reduced to a constant, namely f (n) ≤ 3 2 n + 1.

2005

So far only one approximation algorithm for the number of perfect matchings in general graphs is known. This algorithm of Chien [2] is based on determinants. We present a much simpler algorithm together with some of its variants. One of them has an excellent performance for random graphs, another one might be a candidate for a good worst case performance. We also present an experimental analysis of one of our algorithms.

European Journal of Combinatorics, 2010

We show that every cubic bridgeless graph with n vertices has at least 3n/4 − 10 perfect matchings. This is the first bound that differs by more than a constant from the maximal dimension of the perfect matching polytope. * This research was partially supported by the Czech-Slovenian bilateral project MEB 090805 and BI-CZ/08-09-005.

Journal of Combinatorial Theory, Series B, 1986