# Exponentially many perfect matchings in cubic graphs

@article{Esperet2011ExponentiallyMP, title={Exponentially many perfect matchings in cubic graphs}, author={Louis Esperet and Frantisek Kardos and Andrew D. King and Daniel Kr{\'a}l and Serguei Norine}, journal={Advances in Mathematics}, year={2011}, volume={227}, pages={1646-1664} }

## 49 Citations

### A bound on the number of perfect matchings in Klee-graphs

- MathematicsDiscret. Math. Theor. Comput. Sci.
- 2013

It is shown that every Klee-graph with n ≥8 vertices has at least 3 *2(n+12)/60 perfect matchings, which is close to the bound of Esperet et al. (2011).

### Perfect Matchings in Claw-free Cubic Graphs

- MathematicsElectron. J. Comb.
- 2011

It is proved that every claw-free cubic $n-vertex graph with no cutedge has more than $2^{n/12}$ perfect matchings, thus verifying the conjecture for claw- free graphs.

### Avoiding 5-circuits in a 2-factor of cubic graphs

- MathematicsArXiv
- 2013

It is shown that every bridgeless cubic graph G on n vertices other than the Petersen graph has a 2-factor with at most 2(n−2)/15 circuits of length 5.2, which improves the previously known bound of n/5.83 odd circuits.

### Connected cubic graphs with the maximum number of perfect matchings

- MathematicsJ. Graph Theory
- 2022

It is proved that for n≥6 , the number of perfect matchings in a simple connected cubic graph on 2n vertices is at most 4fn−1 , with fn being the n ‐th Fibonacci number. The unique extremal graph is…

### Upper Bounds for Perfect Matchings in Pfaffian and Planar Graphs

- MathematicsElectron. J. Comb.
- 2013

These upper bounds are better than Bregman's upper bounds on the number of perfect matchings in pfaffian graphs and are applied to fullerene graphs.

### Uniform Generation of d-Factors in Dense Host Graphs

- Computer Science, MathematicsGraphs Comb.
- 2014

Given a constant integer d ≥ 1 and a host graph H that is sufficiently dense, the number of d-factors H contains is lowered and the efficiency of the algorithm provided is justified.

### On the expected number of perfect matchings in cubic planar graphs

- MathematicsTrends in Mathematics
- 2021

This work considers random bridgeless cubic planar graphs with the uniform distribution on graphs with $n$ vertices and shows that the expected number of perfect matchings in labeled bridgless cubicPlanar graphs is asymptotically $c\gamma^n$, where $c>0$ and $\gamma \sim 1.14196$ is an explicit algebraic number.

### Factorially Many Maximum Matchings Close to the Erdős-Gallai Bound

- MathematicsElectron. J. Comb.
- 2022

A classical result of Erdős and Gallai determines the maximum size $m(n,\nu)$ of a graph $G$ of order $n$ and matching number $\nu n$. We show that $G$ has factorially many maximum matchings provided…

### Disjoint odd circuits in a bridgeless cubic graph can be quelled by a single perfect matching

- Mathematics
- 2022

Let G be a bridgeless cubic graph. The Berge–Fulkerson Conjecture (1970s) states that G admits a list of six perfect matchings such that each edge of G belongs to exactly two of these perfect…

## References

SHOWING 1-10 OF 20 REFERENCES

### An improved linear bound on the number of perfect matchings in cubic graphs

- MathematicsEur. J. Comb.
- 2010

### A New Lower Bound on the Number of Perfect Matchings in Cubic Graphs

- MathematicsSIAM J. Discret. Math.
- 2009

We prove that every $n$-vertex cubic bridgeless graph has at least $n/2$ perfect matchings and give a list of all 17 such graphs that have less than $n/2+2$ perfect matchings.

### A superlinear bound on the number of perfect matchings in cubic bridgeless graphs

- MathematicsEur. J. Comb.
- 2012

### Fullerene graphs have exponentially many perfect matchings

- Mathematics
- 2009

A fullerene graph is a planar cubic 3-connected graph with only pentagonal and hexagonal faces. We show that fullerene graphs have exponentially many perfect matchings.

### Explicit construction of regular graphs without small cycles

- MathematicsComb.
- 1984

For every integerd>2 we give an explicit construction of infinitely many Cayley graphsX of degreed withn(X) vertices and girth >0.4801...(logn(X))/log (d−1)−2. This improves a result of Margulis.

### A bound on the number of perfect matchings in Klee-graphs

- MathematicsDiscret. Math. Theor. Comput. Sci.
- 2013

It is shown that every Klee-graph with n ≥8 vertices has at least 3 *2(n+12)/60 perfect matchings, which is close to the bound of Esperet et al. (2011).

### Counting 1-Factors in Regular Bipartite Graphs

- MathematicsJ. Comb. Theory, Ser. B
- 1998

We show that anyk-regular bipartite graph with 2nvertices has at least(k?1)k-1kk-2nperfect matchings (1-factors). Equivalently, this is a lower bound on the permanent of any nonnegative…

### Perfect matchings in planar cubic graphs

- MathematicsComb.
- 2012

It is proved that if G is a planar cubic graph with no cutedge, then G has at least 2-V (G), which is the number of perfect matchings in G.

### Rank of maximum matchings in a graph

- MathematicsMath. Program.
- 1982

The problem of computing the dimension of the face of this polytope which contains the maximum cardinality matchings of a graphG is considered and a good characterization of this quantity is given, in terms of the cyclomatic number of the graph and families of odd subsets of the nodes which are always nearly perfectly matched by every maximum matching.