  We provide a new combinatorial approach to study the minimal free resolutions
of edge ideals, that is, quadratic square-free monomial ideals. With this
method we can recover most of the known results on resolutions of edge ideals
with fuller generality, and at the same time, obtain new results. Past
investigations on the resolutions of edge ideals usually reduced the problem to
computing the dimensions of reduced homology or Koszul homology groups. Our
approach circumvents the highly nontrivial problem of computing the dimensions
of these groups and turns the problem into combinatorial questions about the
associated simple graph. We also show that our technique extends successfully
to the study of graded Betti numbers of arbitrary square-free monomial ideals
viewed as facet ideals of simplicial complexes.
