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

Bounds on the regularity and projective dimension of ideals associated to graphs