Title: Syzygies and Planar Graph Resolution -Abstract- In Algebraic Geometry we can understand the geometry of varieties by studying the properties of the polynomial ring S = k[x_1, ... ,x_r] and its ideals. To effectively study these ideals we need to study also the graded S-modules. The easiest way to describe one of these modules is by giving the generators and the relations among them. The theory of Syzygies and Resolutions arose from this point of view. I will define these objects, and then I will give a combinatorial technique to compute the Syzygies of a S-module coming from a monomial ideal in the trivariate polynomial ring. To a monomial ideal we will associate a graph (the Buchberger Graph) from which we can read all the information needed to describe the S-module associated to the given ideal. Also, this graph allows us to give an efficient algorithm to compute the syzygies, the free resolution, and more. Nothing beyond the basic definition of module will be needed to follow this talk.