41 problems

Let $G$ be a finite simple graph, and let $I_c(G)$ denote its complementary edge ideal in the polynomial ring associated to $G$. Write $\mathcal{R}(I_c(G))$ for the Rees algebra of…

Let $G$ be a graph, let $I(G)$ be its edge ideal, and let ${\rm reg}(I(G))$ denote its Castelnuovo–Mumford regularity. For each integer $s\geq 1$, Alilooee–Banerjee–Beyarslan–Hà's…

Let $\mathcal W_n$ be the wheel graph on $n+1$ vertices, let $I(\mathcal W_n)$ be its edge ideal in $R=\mathbbk[x_1,\ldots,x_n,v]$, and let $\nu(\mathcal W_n)$ denote its matching…

Let $G$ be a simple graph on vertex set $V(G)=[n]$, let $R=\Bbbk[x_1,\ldots,x_n]$, and let $I(G^d)$ be the edge ideal of the $d$-th power of $G$, where $d$ is a positive integer. T…

Let $G$ be a finite simple graph with vertex set $V(G)=\{1,\ldots,n\}$, let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$, and let $I(G)\subseteq S$ be the edge ideal…

Let $G$ be a graph, let $I(G)$ be its edge ideal, and let $I(G)^{(s)}$ denote its $s$-th symbolic power. The invariant ${\rm reg}(-)$ denotes Castelnuovo–Mumford regularity. For ev…

Let ${\Gamma}$ be a connected non-bipartite graph, let $R$ be the polynomial ring associated to its vertices, and let $I$ be the edge ideal of ${\Gamma}$. Depth-drop conjecture. If…

Let ${\Gamma}$ be a connected graph, let $R$ be the polynomial ring associated to its vertices, let $I$ be the edge ideal of ${\Gamma}$, and let ${\mathfrak m}$ be the homogeneous…

Associated-prime extension conjecture. Every associated prime $P$ of $R/(I,f_1,\ldots,f_q)$ is of the form

Let $I({\rm GA}(t,k)')$ be the edge ideal of the complement of the generalized Andrásfai graph ${\rm GA}(t,k)$, and let $\beta_{i,i+j}(I({\rm GA}(t,k)'))$ denote its graded Betti n…

Product bounds conjecture. The following inequalities hold:

Let $G$ be a connected graph, and let $I=I_c(G)$ be its complementary edge ideal. For each $i\geq 0$, let $\HS_i(\mathcal{R}(I))$ denote the $i$-th homological shift ideal of the R…

Symmetric exchange conjecture. The toric ideal

Let $G$ be a chordal graph, let $I(G)$ be its edge ideal, let $I(G)^{[k]}$ denote its $k$-th square-free power, let $\nu(G)$ be its matching number, and let $\mathrm{aim}(G,k)$ be…

Let $G_\omega$ be a weighted tree. Proposition gives necessary conditions for $I(G_\omega)^n$ to be Cohen–Macaulay for all $n\geqslant 1$: $G$ has a perfect matching…

Asymptotic associated-prime conjecture. For all $i\geq 0$,

Let $G$ be a graph with edge ideal $I(G)$. Suppose that, for some integer $q\in\mathbb{N}$, $I(G)^q$ has linear quotients with an ordering $\mathcal N^{(q)}:u_1>\dots>u_r$ of its g…

Linear quotients conjecture for symbolic powers. If $I(G)$ has a linear resolution, then $I(G)^{(k)}$ has linear quotients for every $k\geq 1$.

Componentwise linear symbolic power conjecture. If $I(G)$ has a linear resolution, then $I(G)^{(k)}$ is componentwise linear for every $k\geq 1$.

Let $S$, $I_n$, and $J_n$ be as above, and for each $N$ let $\#\{n\leq N:\,P(n)\}$ denote the number of positive integers $n\leq N$ satisfying $P(n)$. Hilbert-depth frequency conje…

Let $W(C_m)$ be the graph obtained by attaching a whisker to each vertex of the cycle $C_m$, let $I(W(C_m))$ be its edge ideal in the corresponding polynomial ring…

Let $C_n$ be a cycle of length $n$, let $I(C_n)$ be its edge ideal in the corresponding polynomial ring $R=\mathbb{K}[x\mid x\in V(C_n)]$, and let $\nu(C_n)$ denote its matching nu…

Let $G$ be a graph on $m$ vertices and let $\alpha=(\alpha_1,\dots,\alpha_m)\in(\mathbb Z^+)^m$, where $G^\alpha$ is obtained by duplicating the vertices of $G$ according to…

Let $\operatorname{\mathcal{D}}$ be a weighted oriented graph whose underlying graph $G$ is triangle-free, meaning that $G$ contains no cycles of length $3$. The graph $G$ is Cohen…

Restricted bipartite connectivity conjecture. The stable value of the depth of the symbolic powers of $I(G)$ is