Peter Phelan
We provide a characterisation of all graphs whose parity binomial edge ideals have pure resolutions. In particular, we show that the minimal free resolution of a parity binomial edge ideal is pure if and only if the corresponding graph is a complete bipartite graph, or a disjoint union of paths and odd cycles.
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We give an explicit formula for the Hilbert-Poincaré series of the parity binomial edge ideal of a complete graph $K_{n}$ or equivalently for the ideal generated by all $2\times 2$-permanents of a $2\times n$-matrix. It follows that the depth and Castelnuovo-Mumford regularity of these ideals are independent of $n$.
In this paper, we provide exact formulas for the Castelnuovo-Mumford regularity of powers of an almost complete intersection ideal $I$ which is generated by a homogeneous $d$-sequence. As applications, when $I$ is an almost complete intersection, taking the form of the (parity) binomial edge ideal of a connected graph, we can describe explicitly formulas for $\reg(I^t)$ for $t\ge 2$. The only exception is when $I$ is the parity binomial edge ideal of a graph which is obtained by adding an edge between two disjoint odd cycles.
Let $G$ be a simple graph on $n$ vertices. Let $L_G \text{ and } \mathcal{I}_G \: $ denote the Lovász-Saks-Schrijver(LSS) ideal and parity binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1,\ldots, x_n, y_1, \ldots, y_n] $ respectively. We classify graphs whose LSS ideals and parity binomial edge ideals are complete intersections. We also classify graphs whose LSS ideals and parity binomial edge ideals are almost complete intersections, and we prove that their Rees algebra is Cohen-Macaulay. We compute the second graded Betti number and obtain a minimal presentation of LSS i…
Regularity of powers of d-sequence (parity) binomial edge ideals of unicycle graphs
本文分类了单圈图中边缘二项式构成d-序列的图，并研究其（奇偶）二项式边缘理想的幂的正则度。
Let $G$ be a simple graph on $n$ vertices and $\mathcal{I}_G$ denotes parity binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1,\ldots, x_n, y_1, \ldots, y_n].$ We obtain a lower bound for the regularity of parity binomial edge ideals of graphs. We then classify all graphs whose parity binomial edge ideals have regularity $3$. We classify graphs whose parity binomial edge ideals have pure resolution.
Classification of unmixed parity binomial edge ideals of cactus and chordal graphs
本文基于图的结构性质，研究cactus和chordal图的非混杂和Cohen-Macaulay奇偶二项式边理想的分类问题。
Comparison of symbolic and ordinary powers of parity binomial edge ideals
该研究探讨图的奇偶二项式边理想的符号幂与普通幂不等的条件，证明团数大于3、含net或含PT诱导子图时二者存在不相等的幂次，丰富了交换代数与图论交叉领域的相关结论。
Parity binomial edge ideals of simple undirected graphs are introduced. Unlike binomial edge ideals, they do not have square-free Gröbner bases and are radical if only if the graph is bipartite or the characteristic of the ground field is not two. The minimal primes are determined and shown to encode combinatorics of even and odd walks in the graph. A mesoprimary decomposition is determined and shown to be a primary decomposition in characteristic two.
F-Purity of Binomial Edge Ideals
本文研究了二项式边理想F纯性，证明了弱闭图的二项式边理想在特征2时F纯，且反驳了Matsuda的第二个猜想，指出含星形三元组的图在任何正特征下非F纯。
We show that the universal Gröbner basis and the Graver basis of a binomial edge ideal coincide. We provide a description for this basis set in terms of certain paths in the underlying graph. We conjecture a similar result for a parity binomial edge ideal and prove this conjecture for the case when the underlying graph is the complete graph.
In this note, we use estimates given in the recent preprint [Her11b]to deduce a formula for the F-pure threshold of a binomial hypersurface over a field of prime characteristic. These formulas are given in terms of the associated splitting polytope, and remain valid over any characteristic.
We characterize all graphs whose binomial edge ideals have pure resolutions. Moreover, we introduce a new switching of graphs which does not change some algebraic invariants of graphs, and using this, we study the linear strand of the binomial edge ideals for some classes of graphs. Also, we pose a conjecture on the linear strand of such ideals for every graph.
$\mathrm{v}$-number of Lovász-Saks-Schrijver Ideals and (parity) binomial edge ideals of graphs
本文提出计算坐标饱和理想局部v-数的新框架，推导森林图LSS理想局部v-数公式，证明多类图的二项边理想等满足v-数不超过其正则度的结论。
In this article, we obtain an upper bound for the Castelnuovo-Mumford regularity of powers of an ideal generated by a homogeneous quadratic sequence in a polynomial ring in terms of the regularity of its related ideals and degrees of its generators. As a consequence, we compute upper bounds for the regularity of powers of several binomial ideals. We generalize a result of Matsuda and Murai to show that the regularity of $J^s_G$ is bounded below by $2s+\ell(G)-1$ for all $s \geq 1$, where $J_G$ denotes the binomial edge ideal of a graph $G$ and $\ell(G)$ is the length of a longest induced path …
Given a pure binomial ideal I in variables x_i, we define a new measure of the complexity of the saturation of I with respect to the product of the variables x_i, which we call the norm. We give a bound on the norm in terms of easily-computed invariants of the ideal. We discuss statistical applications both practical and theoretical.
Let $G$ be a finite simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in $S = K[x_1, \ldots, x_n, y_1, \ldots, y_n].$ In this article, we prove that if $G$ is a fan graph of a complete graph, then $reg(S/J_G) \leq c(G)$, where $c(G)$ denote the number of maximal cliques in $G$. Further, we show that if $G$ is a $k$-pure fan graph, then $reg(S/J_G) = k+1$. We then compute a precise expression for the regularity of Cohen-Macaulay bipartite graphs.
For a graph $G$, Bolognini et al. have shown $J_{G}$ is strongly unmixed $\Rightarrow$ $J_{G}$ is Cohen-Macaulay $\Rightarrow$ $G$ is accessible, where $J_{G}$ denotes the binomial edge ideals of $G$. Accessible and strongly unmixed properties are purely combinatorial. We give some motivations to focus only on blocks with whiskers for the characterization of all $G$ with Cohen-Macaulay $J_{G}$. We show that accessible and strongly unmixed properties of $G$ depend only on the corresponding properties of its blocks with whiskers and vice versa. Also, we give an infinite class of graphs whose bin…
We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and it has numerous applications within and beyond pure mathematics. The ideals defining toric varieties are precisely the binomial prime ideals. Our main results concern primary decomposition: If $I$ is a binomial ideal then the radical, associated primes, and isolated primary components of $I$ are again binomial, and $I$ admits primary decompositions in terms …
The cut sets of a graph are special sets of vertices whose removal disconnects the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible graphs as the graphs with unmixed binomial edge ideal and whose cut sets form an accessible set system. We prove that the graphs whose binomial edge ideal is Cohen-Macaulay are accessible and we conjecture that the converse holds. We settle the conjecture for large classes of graphs, including chordal and traceable graphs, providing a purely combinatorial d…
Herzog-Hibi-Hreindóttir-Kahle-Rauh introduced the class of closed graph and they proved that the binomial edge ideal $J(G)$ of a graph $G$ has quadratic Gröbner bases if $G$ is closed. In this paper, we introduce the class of weakly closed graph as a generalization of the closed graph and prove that the quotient ring $S/J(G)$ is $F$-pure if $G$ is weakly closed. This fact is a generalization of Ohtani's theorem.
Let $G$ be a simple graph and $I(X_G)=φ^{-1}(x_i^2-x_j^2 : i,j\in V_G)$, where $φ\colon K[E_G]\to K[V_G]$ is the homomorphism that sends an edge to the product of its vertices. The ideal $I(X_G)$ is Cohen--Macaulay, one-dimensional and binomial. If $G$ is bipartite, it is known that the Castelnuovo--Mumford regularity of $I(X_G)$ is equal to the maximum cardinality of a set of edges having no more than half of the edges of any Eulerian subgraph of $G$. Here, with respect to the grevlex order associated to an ordering of the edge set of $G$, we describe a Gröbner basis for $I(X_G)$, and we char…
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