Research ArticlesNo Access

Graded Betti numbers of path ideals of cycles and lines

A. Alilooee

Department of Mathematics, University of Wisconsin-Stout, Menomonie, WI, USA

E-mail Address: alilooeedolataba@uwstout.edu

Corresponding author.

Search for more papers by this author

and

S. Faridi

Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada

E-mail Address: faridi@mathstat.dal.ca

Search for more papers by this author

https://doi.org/10.1142/S0219498818500111Cited by:11 (Source: Crossref)

Abstract

We use purely combinatorial arguments to give a formula to compute all graded Betti numbers of path ideals of paths and cycles. As a consequence, we can give new and short proofs for the known formulas of regularity and projective dimensions of path ideals of paths.

Communicated by B. Buchberger

Keywords:

AMSC: 13A15, 13D02, 05C38, 13F55

References

1. J. Abbott, A. M. Bigatti and G. Lagorio, software. CoCoA-5: a system for doing Computations in Commutative Algebra, http://cocoa.dima.unige.it. Google Scholar
2. A. Alilooee, Algebraic properties of monomial ideals, Ph.D. thesis, Dalhousie University, (2015). Google Scholar
3. A. Alilooee and S. Faridi, On the resolution of path ideals of cycles, Comm. Algebra 43 (2015) 5413–5433. Crossref, Web of Science, Google Scholar
4. R. Bouchat, H. T. Ha and A. O. Keefe, Path ideals of rooted trees and their graded betti numbers, J. Combina. Theory Ser. A 118 (2010) 2411–2425. Crossref, Web of Science, Google Scholar
5. A. Conca and E. D. Negri, M-sequences, graph ideals and ladder ideals of linear types, J. Algebra 211 (1999) 599–624. Crossref, Web of Science, Google Scholar
6. D. R. Grayson and M. E. Stillman, software. Macaulay 2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/. Google Scholar
7. R. Grimaldi, Discrete and Combinatorial Mathematics — An Applied Introduction (3rd edn) (Addison-Wesley, 1993). Google Scholar
8. J. He and A. V. Tuyl, Algebraic properties of path ideal of a tree, Comm. Algebra 38 (2010) 1725–1742. Crossref, Web of Science, Google Scholar
9. S. Jacques, Betti numbers of graph ideals, Ph.D. thesis, The University of Sheffield (2004), arXiv.math. AC/0410107. Google Scholar
10. R. Stanley, Enumerative Combinatorics, Vol. 1, 2nd edn, Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 2012). Google Scholar