On the sequentially Cohen–Macaulay properties of almost complete multipartite graphs

Higashidaira, Hirotaka

doi:10.1080/00927872.2016.1233210

Cited by 5 publications

(4 citation statements)

References 22 publications

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“…By [7] we have the following relations: x 16 x 25 x 15 x 25 x 13 x 25 x 12 x 25 x 10 x 25 x 8 x 25 x 7 x 25 x 6 x 25 x 5 x 25 x 4 x 25 x 3 x 25 x 2 x 25 x 1 x 25…”

Section: Counterexamplesmentioning

confidence: 99%

On vertex decomposability and regularity of graphs

Mafi

Naderi

SOUFIVAND

2023

International Electronic Journal of Algebra

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There are two motivating questions in [M. Mahmoudi, A. Mousivand, M. Crupi, G. Rinaldo, N. Terai and S. Yassemi, arXiv:1006.1087v1] and [M. Mahmoudi, A. Mousivand, M. Crupi, G. Rinaldo, N. Terai and S. Yassemi, J. Pure Appl. Algebra, 215(10) (2011), 2473-2480] about Castelnuovo-Mumford regularity and vertex decomposability of simple graphs. In this paper, we give negative answers to the questions by providing two counterexamples.

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“…By [7] we have the following relations: x 16 x 25 x 15 x 25 x 13 x 25 x 12 x 25 x 10 x 25 x 8 x 25 x 7 x 25 x 6 x 25 x 5 x 25 x 4 x 25 x 3 x 25 x 2 x 25 x 1 x 25…”

Section: Counterexamplesmentioning

confidence: 99%

On vertex decomposability and regularity of graphs

Mafi

Naderi

SOUFIVAND

2023

International Electronic Journal of Algebra

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“…By [16], the regularity of the edge ideal is bounded below by the induced matching number plus 1, and, by [35], it is bounded above by the co-chordal number plus 1. Several classes of graphs for which regularity of the edge ideal attains one, or both, of these bounds have been studied -see [2,11,12,13,14,20,22,33,35]. Refinements of these bounds were recently obtained in [29].…”

Section: Introductionmentioning

confidence: 99%

Joins, Ears and Castelnuovo-Mumford regularity

Neves¹,

Pinto²,

Villarreal³

2019

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We introduce a new class of polynomial ideals associated to a simple graph, G. Let K[E G ] be the polynomial ring on the edges of G and K[V G ] the polynomial ring on the vertices of G. We associate to G an ideal, I(X G ), defined as the preimage of (xwhich sends a variable, te, associated to an edge e = {i, j}, to the product x i x j of the variables associated to its vertices. We show that K[E G ]/I(X G ) is a one-dimensional, Cohen-Macaulay, graded ring, that I(X G ) is a binomial ideal and that, with respect to a fixed monomial order, its initial ideal has a generating set independent of the field K. We focus on the Castelnuovo-Mumford regularity of I(X G ) providing the following sharp upper and lower bounds:where µ(G) is the maximum vertex join number of the graph and b 0 (G) is the number of its connected components. We show that the lower bound is attained for a bipartite graph and use this to derive a new combinatorial result on the number of even length ears of nested ear decomposition.

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“…. , F sr this means that del Ind(C 16 (1,4,8)) (x t ) is pure shellable and Cohen-Macaulay. This is a contradiction by the proof of [2, Theorem 6.1(iii)].…”

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confidence: 97%

“…free vertex decomposable graph; (vii) G is an almost complete multipartite graph such that it is sequentially Cohen-Macaulay or unmixed. For details see[18,9,5,15,13,10,8].…”

mentioning

confidence: 99%