  In this paper we give new upper bounds on the regularity of edge ideals whose
resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the
number of variables. We also give various bounds for the projective dimension
of such ideals, generalizing other recent results. By Alexander duality, our
results also apply to unmixed square-free monomial ideals of codimension two.
We also discuss and connect these results to more classical topics in
commutative algebra.
