69 problems

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…

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 with edge ideal $I(G)$, and let $I(G)^{(t)}$ denote its $t$-th symbolic power. The Castelnuovo–Mumford regularity of an ideal is denoted by $\operatorname{reg}$.…

Eventual homological linearity conjecture. If $I(G)$ has a linear resolution, then $\operatorname{HS}_i(I(G)^k)$ has a linear resolution for all $i\geq 0$ and all $k\gg 0$.

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 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…

Eliahou–Villarreal conjecture. If the edge ideal of a connected graph $G$ has a 2-linear resolution, then

Banerjee–Beyarslan–Hà conjecture.

Let $T$ be a forest, let $I(T)$ be its edge ideal, and let a tree-like system for $I(T)$ be a system whose support is the set of edge monomials of $I(T)$. For stretched forests, th…

Let $R$ be a polynomial ring over a field whose indeterminates are the vertices of a forest $G$, and let $I(G)$ be the edge ideal generated by the squarefree quadratic monomials co…

Let $G$ be a graph, let $I(G)$ be its edge ideal, and let $I(G)^{[p]}$ denote the ideal generated by the squarefree monomials among the generators of $I(G)^p$. Write $\nu_0(G)$ for…

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

Let $G$ be a forest, let $I(G)$ be its edge ideal, and for $q\geq 1$ let $I(G)^{[q]}$ be the ideal generated by squarefree monomials corresponding to matchings of size $q$ in $G$.…

Let $C_n$ be a cycle on $n$ vertices, let $W(C_n)$ denote its whiskered cycle, and let $I(G)$ be the edge ideal of a finite simple graph $G$. For $q\geq 1$, let $I(G)^{[q]}$ be the…

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 $G$ be a forest, 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 the max…

Hà–Katsabekis–Uliczka conjecture. The converse inclusion also holds; equivalently,

Let $L_n$ be the path graph on $n$ vertices, let $I=I(L_n)$ be its edge ideal, and let $I^{[k]}$ denote its $k$-th square-free power. Write $\mathrm{match}(L_n)$ for the matching n…

Let $G$ be a line graph, let $I(G)$ denote its edge ideal, and let $I(G)^{[k]}$ denote its $k$-th square-free power. Conjecture. The v-number of the ordinary powers of the edge ide…

Characterization conjecture. An edge ideal $I(G)$ has homological linear quotients if and only if $G$ is co-chordal and $\operatorname{H}_n^c$-free for any $n\geq 6$.

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…