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

2019 ◽  
Vol 27 (3) ◽  
pp. 113-135
Author(s):  
Zahid Iqbal ◽  
Muhammad Ishaq
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Quotient Ring ◽  
Edge Ideal ◽  
Stanley Depth ◽  
Edge Ideals

AbstractLet 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.

scholarly journals Stanley depth of edge ideals

2012 ◽  
Vol 49 (4) ◽  
pp. 501-508 ◽  
Author(s):  
Muhammad Ishaq ◽  
Muhammad Qureshi
Keyword(s):  
Complete Graph ◽  
Upper Bound ◽  
Edge Ideal ◽  
Stanley Depth ◽  
Edge Ideals

We give an upper bound for the Stanley depth of the edge ideal I of a k-partite complete graph and show that Stanley’s conjecture holds for I. Also we give an upper bound for the Stanley depth of the edge ideal of a s-uniform complete bipartite hypergraph.

scholarly journals Squarefree Powers of Edge Ideals of Forests

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Nursel Erey ◽  
Takayuki Hibi
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Matching Number ◽  
Edge Ideal ◽  
Combinatorial Formula ◽  
Edge Ideals ◽  
Linear Resolution

Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive integer $k$, we classify all forests $G$ such that $I(G)^{[k]}$ has linear resolution. We also give a combinatorial formula for the regularity of $I(G)^{[2]}$ for any forest $G$.

scholarly journals Weakly Closed Graphs and F-Purity of Binomial Edge Ideals

2018 ◽  
Vol 25 (04) ◽  
pp. 567-578
Author(s):  
Kazunori Matsuda
Keyword(s):  
Polynomial Ring ◽  
Quotient Ring ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Edge Ideal ◽  
Closed Graph ◽  
Edge Ideals ◽  
Weakly Closed

Herzog, Hibi, Hreindóttir et al. introduced the class of closed graphs, and they proved that the binomial edge ideal JG of a graph G has quadratic Gröbner bases if G is closed. In this paper, we introduce the class of weakly closed graphs as a generalization of the closed graph, and we prove that the quotient ring S/JG of the polynomial ring [Formula: see text] with K a field and [Formula: see text] is F-pure if G is weakly closed. This fact is a generalization of Ohtani’s theorem.

scholarly journals On the Betti Numbers of some Classes of Binomial Edge Ideals

10.37236/3689 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Sohail Zafar ◽  
Zohaib Zahid
Keyword(s):  
Lower Bounds ◽  
Betti Numbers ◽  
Edge Ideal ◽  
Induced Subgraphs ◽  
Edge Ideals ◽  
Arbitrary Graphs

We study the Betti numbers of binomial edge ideal associated to some classes of graphs with large Castelnuovo-Mumford regularity. As an application we give several lower bounds of the Castelnuovo-Mumford regularity of arbitrary graphs depending on induced subgraphs.

scholarly journals Comparing powers of edge ideals

2019 ◽  
Vol 18 (10) ◽  
pp. 1950184 ◽  
Author(s):  
Mike Janssen ◽  
Thomas Kamp ◽  
Jason Vander Woude
Keyword(s):  
Symbolic Power ◽  
Edge Ideal ◽  
Homogeneous Ideal ◽  
Edge Ideals ◽  
Ideal Formula ◽  
Number Of Generators ◽  
Recent Interest ◽  
Odd Cycle ◽  
Ordinary Power

Given a nontrivial homogeneous ideal [Formula: see text], a problem of great recent interest has been the comparison of the [Formula: see text]th ordinary power of [Formula: see text] and the [Formula: see text]th symbolic power [Formula: see text]. This comparison has been undertaken directly via an exploration of which exponents [Formula: see text] and [Formula: see text] guarantee the subset containment [Formula: see text] and asymptotically via a computation of the resurgence [Formula: see text], a number for which any [Formula: see text] guarantees [Formula: see text]. Recently, a third quantity, the symbolic defect, was introduced; as [Formula: see text], the symbolic defect is the minimal number of generators required to add to [Formula: see text] in order to get [Formula: see text]. We consider these various means of comparison when [Formula: see text] is the edge ideal of certain graphs by describing an ideal [Formula: see text] for which [Formula: see text]. When [Formula: see text] is the edge ideal of an odd cycle, our description of the structure of [Formula: see text] yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.

scholarly journals A Turán Type Problem Concerning the Powers of the Degrees of a Graph

10.37236/1525 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Degree Sequence ◽  
Upper And Lower Bounds ◽  
Exact Results ◽  
Type Problem ◽  
Maximum Value ◽  
Turán Number

For a graph $G$ whose degree sequence is $d_{1},\ldots ,d_{n}$, and for a positive integer $p$, let $e_{p}(G)=\sum_{i=1}^{n}d_{i}^{p}$. For a fixed graph $H$, let $t_{p}(n,H)$ denote the maximum value of $e_{p}(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $t_{1}(n,H)$ is twice the Turán number of $H$. In this paper we consider the case $p>1$. For some graphs $H$ we obtain exact results, for some others we can obtain asymptotically tight upper and lower bounds, and many interesting cases remain open.

scholarly journals Stanley depth of squarefree Veronese ideals

2013 ◽  
Vol 21 (3) ◽  
pp. 67-72 ◽  
Author(s):  
Mircea Cimpoeas
Keyword(s):  
Quotient Ring ◽  
Stanley Depth

Abstract We compute the Stanley depth for the quotient ring of a square free Veronese ideal and we give some bounds for the Stanley depth of a square free Veronese ideal. In particular, it follows that both satisfy the Stanley's conjecture.

scholarly journals Edge ideals of clique clutters of comparability graphs and the normality of monomial ideals

2010 ◽  
Vol 106 (1) ◽  
pp. 88 ◽  
Author(s):  
Luis A. Dupont ◽  
Rafael H. Villarreal
Keyword(s):  
Monomial Ideal ◽  
Lattice Points ◽  
Edge Ideal ◽  
Comparability Graph ◽  
Decomposition Property ◽  
Edge Ideals ◽  
Finite Poset ◽  
Integer Decomposition Property ◽  
Torsion Free ◽  
Min Cut

The normality of a monomial ideal is expressed in terms of lattice points of blocking polyhedra and the integer decomposition property. For edge ideals of clutters this property characterizes normality. Let $G$ be the comparability graph of a finite poset. If $\mathrm{cl}(G)$ is the clutter of maximal cliques of $G$, we prove that $\mathrm{cl}(G)$ satisfies the max-flow min-cut property and that its edge ideal is normally torsion free. Then we prove that edge ideals of complete admissible uniform clutters are normally torsion free.

scholarly journals Further results on a generalization of Bertrand's postulate

1996 ◽  
Vol 19 (1) ◽  
pp. 75-85
Author(s):  
George Giordano
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Bertrand’S Postulate

Letd(k)be defined as the least positive integernfor whichpn+1<2pn−k. In this paper we will show that fork≥286664, thend(k)<k/(logk−2.531)and fork≥2, thenk(1−1/logk)/logk<d(k). Furthermore, forksufficiently large we establish upper and lower bounds ford(k).

A Note on the Multi-Colored Bipartite Ramsey Number for Paths

2021 ◽  
pp. 2150041
Author(s):  
Hanxiao Qiao ◽  
Ke Wang ◽  
Suonan Renqian ◽  
Renqingcuo
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Bipartite Graphs ◽  
Ramsey Number ◽  
Upper And Lower Bounds ◽  
Number Formula

For bipartite graphs [Formula: see text], the bipartite Ramsey number [Formula: see text] is the least positive integer [Formula: see text] so that any coloring of the edges of [Formula: see text] with [Formula: see text] colors will result in a copy of [Formula: see text] in the [Formula: see text]th color for some [Formula: see text]. In this paper, we get the exact value of [Formula: see text], and obtain the upper and lower bounds of [Formula: see text], where [Formula: see text] denotes a path with [Formula: see text] vertices.

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