The Lovász-Plummer conjecture asserts that the number 
of perfect matchings
of a finite graph 
that is cubic and has no bridges is exponential in its vertex
count 
.
More precisely, there is an absolute constant 
such that
 |
for every such graph (Lovász and Plummer 1986).
Esperet et al. (2011) proved the conjecture by establishing
 |
This is a sufficient bound, not a claim that the constant 
is optimal. The published theorem
also allows multiple edges but excludes graph
loops. Graphs need not be connected, since
the numbers of perfect matchings multiply over
the connected components. The result strengthens
the existence conclusion of Petersen's theorem
to an exponential lower bound.
