Vol. 9, No. 5, 2016

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Splitting techniques and Betti numbers of secant powers

Reza Akhtar, Brittany Burns, Haley Dohrmann, Hannah Hoganson, Ola Sobieska and Zerotti Woods

Vol. 9 (2016), No. 5, 737–750
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.

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Keywords
Betti number, edge ideal, secant power, complete graph, complete bipartite graph
Mathematical Subject Classification 2010
Primary: 13D02
Secondary: 05C25
Milestones
Received: 31 December 2014
Revised: 23 July 2015
Accepted: 27 October 2015
Published: 25 August 2016

Communicated by Scott T. Chapman
Authors
Reza Akhtar
Department of Mathematics
Miami University
Oxford, OH 45056
United States
Brittany Burns
Department of Mathematics
University of Central Florida
Orlando, FL 32816
United States
Haley Dohrmann
Department of Mathematics
University of Iowa
Iowa City, IA 52242
%-1419
United States
Hannah Hoganson
Department of Mathematics
University of Utah
Salt Lake City, UT 84112
%-0090
United States
Ola Sobieska
Department of Mathematics
Texas A&M University
College Station, TX 77843
%-3368
United States
Zerotti Woods
Department of Mathematics
University of Georgia
Athens, GA 30602
United States