On the resolutions of the edge ideals of graphs

Margherita Barile

doi:10.59400/jam2176

On the resolutions of the edge ideals of graphs

Margherita Barile Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, 70125 Bari, Italy

Article ID: 2176

DOI: https://doi.org/10.59400/jam2176

Keywords: minimal graded free resolutions; cellular resolutions; Morse theory; edge ideals of graphs; monomial ideals

Abstract

The so-called bridge-friendliness is a set of divisibility conditions on the minimal generators of monomial ideals. It was introduced by Chau and Kara as a sufficient criterion for the existence of a cellular minimal graded free resolution. It is fulfilled by large classes of monomial ideals, in particular by the edge ideals of acyclic graphs. We present a construction that, given a pair of graphs with bridge-friendly edge ideals, produces a new graph with the same property. An additional assumption is that the starting graphs both have at least one leaf.

References

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[2]Mermin J. Three simplicial resolutions. In: Francisco C, Klingler LC, Sather-Wagstaff S, Vassilev JC (editors). Progress in Commutative Algebra 1. De Gruyter; 2012. pp. 127–141.

[3]Barile M, Macchia A. Minimal cellular resolutions of the edge ideals of forests. The Electronic Journal of Combinatorics. 2020; 27(2). doi: 10.37236/8810

[4]Chau T, Hà TH, Maithani A. Powers of edge ideals of graphs with minimal cellular resolutions. Available online: https://arxiv.org/html/2404.04380v1 (accessed on 20 November 2024).

[5]Faridi S, Hà TH, Hibi T, Morey S. Scarf complexes of graphs and their powers. Available online: https://arxiv.org/abs/2403.05439 (accessed on 20 November 2024).

[6]Chau T, Kara S, Wang K. Minimal cellular resolutions of path ideals. Available online: https://arxiv.org/html/2403.16324v1 (accessed on 20 November 2024).

[7]Batzies E, Welker W. Discrete Morse theory for cellular resolutions. Journal für die reine und angewandte Mathematik. 2002(543): 147–168. doi: 10.1515/crll.2002.012

[8]Chau T, Kara S. Barile-Macchia resolutions. Journal of Algebraic Combinatorics. 2024; 59(2): 413–472. doi: 10.1007/s10801-023-01293-9

Published

2025-03-26

How to Cite

Barile, M. (2025). On the resolutions of the edge ideals of graphs. Journal of AppliedMath, 3(2), 2176. https://doi.org/10.59400/jam2176

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