  We solve the monomer-dimer problem on a non-bipartite lattice, the simple
quartic lattice with cylindrical boundary conditions, with a single monomer
residing on the boundary. Due to the non-bipartite nature of the lattice, the
well-known method of a Temperley bijection of solving single-monomer problems
cannot be used. In this paper we derive the solution by mapping the problem
onto one on close-packed dimers on a related lattice. Finite-size analysis of
the solution is carried out. We find from asymptotic expansions of the free
energy that the central charge in the logarithmic conformal field theory
assumes the value $c=-2$.
