  We study the defining equations of the Rees algebra of square-free monomial
ideals in a polynomial ring over a field. We determine that when an ideal $I$
is generated by $n$ square-free monomials of the same degree then $I$ has
relation type at most $n-2$ as long as $n \leq 5$. In general, we establish the
defining equations of the Rees algebra in this case. Furthermore, we give a
class of examples with relation type at least $n-2$, where $n=\mu(I)$. We also
provide new classes of ideals of linear type. We propose the construction of a
graph, namely the generator graph of an ideal, where the monomial generators
serve as vertices for the graph. We show that when $I$ is a square-free
monomial ideal generated in the same degree that is at least 2 and the
generator graph of $I$ is the graph of a disjoint union of trees and graphs
with unique odd cycles then $I$ is an ideal of linear type.
