Splitting techniques and Betti numbers of secant powers

Abstract

Using ideal-splitting techniques, we prove a recursive formula relating the Betti numbers of the secant powers of the edge ideal of a graph $H$ to those of the join of $H$ with a finite independent set. We apply this result in conjunction with other splitting techniques to compute these Betti numbers for wheels, complete graphs and complete multipartite graphs, recovering and extending some known results about edge ideals.