31 problems

Let $G$ be a graph on $n\ge 7$ non-isolated vertices. Suppose that $\operatorname{reg}(J_G)=n-1$. Projective-dimension bound. Then … The paper reports that this stronger bound is s…

Let $C_n$ and $C_m$ be cycles with $n,m\geq 3$, let $J_{C_n,C_m}$ be their associated binomial edge ideal, and let $\ell(J_{C_n,C_m})$ denote its analytic spread. Assume that the c…

Let $T_1$ and $T_2$ be trees on $n$ and $m$ vertices, respectively, and fix $i\in V(T_1)$ and $j\in V(T_2)$. For a positive integer $l$, define $G_1$ as the tree on $n+m+l$ vertice…

Let $G$ and $H$ be graphs on $n$ and $m$ vertices, respectively, with $n,m\geq 3$. Let $J_{G,H}$ be the associated binomial edge ideal, let $\ell(J_{G,H})$ denote its analytic spre…

Let $G$ be a graph, let $\Bbbk$ be a field, and let $R_G$ denote the quotient by the binomial edge ideal associated to $G$. Write $p$ for the characteristic of the base field. Mats…

Let $G$ be a simple graph on $n$ vertices, let $\Bbbk$ be a field, and set $R=\Bbbk[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$. For $\{i,j\}\in E(G)$, write…

Let $G$ be an enumerated graph. Let $\ell$ be the length of the longest induced path in $G$, and let $\ell_{<}$ be the length of the longest admissible path in $G$. Write…

Let $N$ be the net graph. Its binomial edge ideal is denoted by $J_N$, and write $\widehat{\operatorname{reg}}(J_N)$ for its asymptotic regularity. Net graph asymptotic regularity…

Let $G$ be a finite simple graph, with $n(G)$ denoting its number of vertices and $J_G$ its binomial edge ideal in $S=K[x_i,y_i:i\in V(G)]$. Matsuda–Murai's conjecture. … if and on…

Let $G=C_{n,n}$ with $n \ge 4$. The graph $G$ is the crown graph obtained from the complete bipartite graph on two parts of size $n$ by deleting a perfect matching, and $J_G$ denot…

Bolognini–Macchia–Strazzanti's conjecture. $J_G$ is Cohen–Macaulay if and only if $G$ is accessible.

Let $G$ be a graph, let $S=\mathbb{K}[x_1,\dots,x_n,y_1,\dots,y_n]$, and let … be its binomial edge ideal. An ideal $I\subset A$ is of linear type when its symmetric algebra is nat…

Conjecture for cycles and binary trees. The following equalities hold:

Let $G$ be a finite simple graph on vertex set $[n]$. For $S\subseteq[n]$, let $c_G(S)$ be the number of connected components of $G-S$, and call $S$ a cut set if…

Let $G$ be a graph, and let its binomial edge ideal be the ideal generated by the binomials associated to the edges of $G$. A graph is unicyclic if it has exactly one cycle. Jayant…

Let $G$ be a graph, let $J_G$ be its binomial edge ideal, and say that $J_G$ is of linear type when its Rees algebra is generated by the degree-one part in the standard sense. Line…

Let $G$ be a graph on $n$ vertices, let $J_G$ be its binomial edge ideal in a polynomial ring $S$, and let $h_{S/J_G}(t)$ be the numerator polynomial of the Hilbert series of…

Let $G$ be a simple graph, let $S$ be the polynomial ring used to define its binomial edge ideal $J_G$, and let $c(G)$ be the number of maximal cliques of $G$. Madani–Kiani's conje…

Let $G$ be a closed graph, let $J_G$ be its binomial edge ideal, and let $\ini_<(J_G)$ denote its initial ideal with respect to the specified term order. For every integer…

The clique-count regularity conjecture. For every graph $G$,

Let $G$ be a graph on $[n]$, let $S=K[x_1,\ldots,x_n,y_1,\ldots,y_n]$, and let $J_G$ be its binomial edge ideal. Let $c(G)$ denote the number of maximal cliques in $G$. Saeedi Mada…

Let $G$ be a finite simple graph, let $J_G$ denote its binomial edge ideal, and let $C_n$ denote the cycle graph on $n$ vertices. Let $\mathcal{R}(J_G)$ be the Rees algebra of…

Let $A=\mathbb{K}[x_1,\dots,x_n]$ be a polynomial ring over a field $\mathbb{K}$, let $I\subseteq A$ be a $\mathbb{Z}^m$-graded Cartwright–Sturmfels ideal, and let…

Let $G$ be a graph, let $S={\mathbb K}[x_1,\ldots,x_n,y_1,\ldots,y_n]$, and let $J_G$ be its binomial edge ideal. Denote by $c(G)$ the number of maximal cliques of $G$. Saeedi and…

Let $G$ be a finite simple graph on the vertex set $[n]=\{1,\ldots,n\}$, let $K[\boldsymbol{x},\boldsymbol{y}]=K[x_1,\ldots,x_n,y_1,\ldots,y_n]$ be the standard graded polynomial r…