56 problems

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 $G$ 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 closed graph, let $J_G$ be its binomial edge ideal, and let $<$ be a diagonal term order. Ene–Herzog–Hibi conjecture. The graded Betti numbers of $J_G$ agree with thos…

Let $G$ be a graph, let $S$ be the standard graded polynomial ring associated with $G$, and let $J_G$ be its binomial edge ideal. Write $\mathrm{reg}(S/J_G)$ for the Castelnuovo–Mu…

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…

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

The AT-free graph conjecture. For any AT-free graph $G$, $\mathcal{J}(G)$ is of König type.

Let $G$ be an arbitrary graph, let $J_G$ denote its binomial edge ideal, and say that $G$ is accessible in the sense of Bolognini, Macchia and Strazzanti. Bolognini–Macchia–Strazza…

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 simple graph on the vertex set $[n]$, let $S=K[x_1,\ldots,x_n,y_1,\ldots,y_n]$, and let $J_G$ be its binomial edge ideal. Fix the specified monomial order $<$ and writ…

Let $G$ be a graph. Write $J_G$ for its binomial edge ideal, let $\beta_{i,j}(J_G)$ denote its graded Betti numbers, and let $k_r(G)$ denote the number of $r$-cliques of $G$. Conje…

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 $G$ 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 a closed graph, let $J_G$ be its binomial edge ideal, and let $\operatorname{in}_{\operatorname{lex}}(J_G^m)$ denote the lexicographic initial ideal of $J_G^m$. 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 $C_n$ be the cycle graph on $n$ vertices, and let $J_{C_n}$ denote its binomial edge ideal. The v-number of a homogeneous ideal is the minimum degree of a homogeneous polynomia…

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…

Dey's conjecture. For $n\geq 6$, equality holds in the bound

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: