(S2)-condition and Cohen–Macaulay binomial edge ideals
We describe the simplicial complex Δ such that the initial ideal of the binomial edge ideal J G of G is the Stanley-Reisner ideal of Δ . By using Δ we show that if J G is ( S 2 ) , then G is accessible. We also characterize all accessible blocks with whiskers of cycle rank 3 and we define a new infi...
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| Published in | Journal of algebraic combinatorics Vol. 57; no. 2; pp. 589 - 615 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.03.2023
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0925-9899 1572-9192 1572-9192 |
| DOI | 10.1007/s10801-022-01173-8 |
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| Abstract | We describe the simplicial complex
Δ
such that the initial ideal of the binomial edge ideal
J
G
of G is the Stanley-Reisner ideal of
Δ
. By using
Δ
we show that if
J
G
is
(
S
2
)
, then G is accessible. We also characterize all accessible blocks with whiskers of cycle rank 3 and we define a new infinite class of accessible blocks with whiskers for any cycle rank. Finally, by using a computational approach, we show that the graphs with at most 12 vertices whose binomial edge ideal is Cohen–Macaulay are all and only the accessible ones. |
|---|---|
| AbstractList | We describe the simplicial complex
$$\Delta $$
Δ
such that the initial ideal of the binomial edge ideal
$$J_\textrm{G}$$
J
G
of G is the Stanley-Reisner ideal of
$$\Delta $$
Δ
. By using
$$\Delta $$
Δ
we show that if
$$J_\textrm{G}$$
J
G
is
$$(S_2)$$
(
S
2
)
, then G is accessible. We also characterize all accessible blocks with whiskers of cycle rank 3 and we define a new infinite class of accessible blocks with whiskers for any cycle rank. Finally, by using a computational approach, we show that the graphs with at most 12 vertices whose binomial edge ideal is Cohen–Macaulay are all and only the accessible ones. We describe the simplicial complex Δ such that the initial ideal of the binomial edge ideal J G of G is the Stanley-Reisner ideal of Δ . By using Δ we show that if J G is ( S 2 ) , then G is accessible. We also characterize all accessible blocks with whiskers of cycle rank 3 and we define a new infinite class of accessible blocks with whiskers for any cycle rank. Finally, by using a computational approach, we show that the graphs with at most 12 vertices whose binomial edge ideal is Cohen–Macaulay are all and only the accessible ones. We describe the simplicial complex Δ such that the initial ideal of the binomial edge ideal JG of G is the Stanley-Reisner ideal of Δ. By using Δ we show that if JG is (S2), then G is accessible. We also characterize all accessible blocks with whiskers of cycle rank 3 and we define a new infinite class of accessible blocks with whiskers for any cycle rank. Finally, by using a computational approach, we show that the graphs with at most 12 vertices whose binomial edge ideal is Cohen–Macaulay are all and only the accessible ones. |
| Author | Lerda, Alberto Mascia, Carla Rinaldo, Giancarlo Romeo, Francesco |
| Author_xml | – sequence: 1 givenname: Alberto surname: Lerda fullname: Lerda, Alberto organization: University of Trento – sequence: 2 givenname: Carla surname: Mascia fullname: Mascia, Carla organization: University of Trento – sequence: 3 givenname: Giancarlo orcidid: 0000-0003-3644-1142 surname: Rinaldo fullname: Rinaldo, Giancarlo email: giancarlo.rinaldo@unitn.it organization: University of Trento – sequence: 4 givenname: Francesco surname: Romeo fullname: Romeo, Francesco organization: University of Trento |
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| CitedBy_id | crossref_primary_10_1016_j_jalgebra_2023_05_037 crossref_primary_10_1016_j_ejc_2025_104123 crossref_primary_10_1007_s40306_024_00540_w crossref_primary_10_1007_s00009_024_02685_2 |
| Cites_doi | 10.4171/JCA/2-3-2 10.1007/s13348-019-00268-z 10.1142/S0219498819500725 10.1016/j.ejc.2017.11.004 10.1080/00927870903527584 10.1215/00277630-1431831 10.21236/AD0705364 10.1016/j.aam.2010.01.003 10.1080/00927872.2012.709569 10.1080/00927872.2019.1596278 10.1080/00927872.2014.952734 10.1090/proc/13241 10.1016/j.jcta.2020.105278 10.1007/s10801-021-01088-w 10.1007/s00222-020-00958-7 |
| ContentType | Journal Article |
| Copyright | The Author(s) 2022 The Author(s) 2022. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| DOI | 10.1007/s10801-022-01173-8 |
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| Keywords | 05E40 Serre’s condition Cohen–Macaulay rings 13C05 05C25 13C14 Binomial edge ideals Accessible chain of cycles |
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| References | CR2 CR4 Banerjee, Núñez-Betancourt (CR1) 2017; 145 Ene, Rinaldo, Terai (CR7) 2020; 175 Conca, De Negri, Gorla (CR5) 2018; 2 Kiani, Madani (CR11) 2015; 43 Terai (CR19) 2007 Bolognini, Macchia, Strazzanti (CR3) 2018; 70 CR13 CR12 CR21 Villarreal (CR20) 2001 Ene, Herzog, Hibi (CR6) 2011; 45 Montaner (CR14) 2020; 71.2 Harary (CR8) 1969 Ohtani (CR15) 2011; 39 Rinaldo (CR17) 2013; 56(104) Jayanthan, Kumar (CR10) 2019; 47 Rinaldo (CR18) 2019; 18 Herzog, Hibi, Hreinsdóttir, Kahle, Rauh (CR9) 2010; 45 Rauf, Rinaldo (CR16) 2014; 42 1173_CR2 1173_CR4 A Conca (1173_CR5) 2018; 2 A Banerjee (1173_CR1) 2017; 145 RH Villarreal (1173_CR20) 2001 N Terai (1173_CR19) 2007 F Harary (1173_CR8) 1969 J.Á. Montaner (1173_CR14) 2020; 71.2 G Rinaldo (1173_CR17) 2013; 56(104) V Ene (1173_CR7) 2020; 175 J Herzog (1173_CR9) 2010; 45 D Kiani (1173_CR11) 2015; 43 V Ene (1173_CR6) 2011; 45 D Bolognini (1173_CR3) 2018; 70 A Rauf (1173_CR16) 2014; 42 G Rinaldo (1173_CR18) 2019; 18 1173_CR12 1173_CR21 AV Jayanthan (1173_CR10) 2019; 47 1173_CR13 M Ohtani (1173_CR15) 2011; 39 |
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| Snippet | We describe the simplicial complex
Δ
such that the initial ideal of the binomial edge ideal
J
G
of G is the Stanley-Reisner ideal of
Δ
. By using
Δ
we show... We describe the simplicial complex $$\Delta $$ Δ such that the initial ideal of the binomial edge ideal $$J_\textrm{G}$$ J G of G is the Stanley-Reisner ideal... We describe the simplicial complex Δ such that the initial ideal of the binomial edge ideal JG of G is the Stanley-Reisner ideal of Δ. By using Δ we show that... |
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| SubjectTerms | Accessibility Algebra Apexes Combinatorics Computer Science Convex and Discrete Geometry Decomposition Graph theory Graphs Group Theory and Generalizations Lattices Mathematics Mathematics and Statistics Order Ordered Algebraic Structures |
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| Title | (S2)-condition and Cohen–Macaulay binomial edge ideals |
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