<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>k</italic> be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the <italic>kth</italic> power of a path on <italic>n</italic> vertices. We show that both depth and Stanley depth have the same values and can be given in terms of <italic>k</italic> and <italic>n</italic>. If <italic>n</italic>≣0, <italic>k</italic> + 1, <italic>k</italic> + 2, . . . , 2<italic>k</italic>(mod(2<italic>k</italic> + 1)), then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the <italic>kth</italic> power of a cycle on <italic>n</italic> vertices and tight bounds otherwise, in terms of <italic>n</italic> and <italic>k</italic>. We also compute lower bounds for the Stanley depth of the edge ideals associated to the <italic>kth</italic> power of a path and a cycle and prove a conjecture of Herzog for these ideals.</abstract>

Let k be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a path on n vertices. We show that both depth and Stanley depth have the same values and can be given in terms of k and n. If n≣0, k + 1, k + 2, . . . , 2k(mod(2k + 1)), then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a cycle on n vertices and tight bounds otherwise, in terms of n and k. We also compute lower bounds for the Stanley depth of the edge ideals associated to the kth power of a path and a cycle and prove a conjecture of Herzog for these ideals.

Depth and Stanley depth of the edge ideals of the powers of paths and cycles

National University of Sciences and Technology Islamabad

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Depth and Stanley depth of the edge ideals of the powers of paths and cycles"}

Let k be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a path on n vertices. We...

Depth and Stanley depth of the edge ideals of the powers of paths and cycles

Published Online: Dec 21, 2019

Page range: 113 - 135

Received: Nov 06, 2018

Accepted: Jan 11, 2019

DOI: https://doi.org/10.2478/auom-2019-0037

Keywords
Monomial ideal, edge ideal, depth, Stanley decomposition, Stanley depth

© 2019 Zahid Iqbal et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Depth and Stanley depth of the edge ideals of the powers of paths and cycles

Published Online: Dec 21, 2019

Page range: 113 - 135

Received: Nov 06, 2018

Accepted: Jan 11, 2019

DOI: https://doi.org/10.2478/auom-2019-0037

KeywordsMonomial ideal, edge ideal, depth, Stanley decomposition, Stanley depth

© 2019 Zahid Iqbal et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Monomial ideal, edge ideal, depth, Stanley decomposition, Stanley depth