5 problems

Let $G$ be a triangular cactus graph, meaning a cactus graph whose blocks are all $3$-cycles, and let its diameter be the maximum distance between two vertices of $G$. The $(S_2)$-…

Let $G=(V,E)$ be a graph, let $n=|V|$, and let $k[G]=k[x_1,\ldots,x_n]/I(G)$ be its edge ring, where $I(G)$ is generated by $\{x_ix_j:\{i,j\}\in E\}$. Write…

Let $G$ be a finite connected simple graph, and let $K[G]$ denote its edge ring. For an integer $q\geq 4$, assume that $K[G]$ has a $q$-linear resolution. Hibi–Matsuda–Tsuchiya con…

Let $G$ be a finite connected simple graph, and let $K[G]$ denote its edge ring. A graded ring has a $q$-linear resolution when its minimal graded free resolution is $q$-linear, an…

Let $G$ be a finite graph on the vertex set $[d]=\{1,\ldots,d\}$ with $d\geq 7$, and let $K[G]$ be its edge ring. Depth-seven conjecture. … The claim arises from computations showi…