  An important development in the field of the fractional quantum Hall effect
has been the proposal that the 5/2 state observed in the Landau level with
orbital index $n = 1$ of two dimensional electrons in a GaAs quantum well
originates from a chiral $p$-wave paired state of composite fermions which are
topological bound states of electrons and quantized vortices. This state is
theoretically described by a "Pfaffian" wave function or its hole partner
called the anti-Pfaffian, whose excitations are neither fermions nor bosons but
Majorana quasiparticles obeying non-Abelian braid statistics. This has inspired
ideas on fault-tolerant topological quantum computation and has also instigated
a search for other states with exotic quasiparticles. Here we report
experiments on monolayer graphene that show clear evidence for unexpected
even-denominator fractional quantum Hall physics in the $n=3$ Landau level. We
numerically investigate the known candidate states for the even-denominator
fractional quantum Hall effect, including the Pfaffian, the particle-hole
symmetric Pfaffian, and the 221-parton states, and conclude that, among these,
the 221-parton appears a potentially suitable candidate to describe the
experimentally observed state. Like the Pfaffian, this state is believed to
harbour quasi-particles with non-Abelian braid statistics
