8 problems

De Klerk–Pasechnik conjecture. For every graph $G$,

de Klerk's conjecture. For every graph $G$,

For $k\geq 1$, let $G_k$ be the graph obtained from the complete bipartite graph $K_{k+1,k+1}$ with bipartition $U=\{u_0,\ldots,u_k\}$ and $V=\{v_0,\ldots,v_k\}$ by subdividing eac…

Let $G$ be a graph, and let $G^\bullet$ be the graph obtained by adding an isolated vertex to $G$. Define … where $\nu^{(r)}(G)$ is the copositive-based hierarchy using the cones…

Let $G=(V,E)$ be a graph, let $\vartheta^{(r)}(G)$ be its order-$r$ copositive hierarchy bound, and let $S^\perp$ denote the vertices adjacent to or equal to a stable set $S$. For…

Let $G$ be a graph and let $G\oplus i_0$ be the graph obtained by adding an isolated vertex $i_0$. The isolated-node monotonicity conjecture. … Thus, adding an isolated vertex shou…

Let $n\geq1$, let $\mathcal S^n$ be the space of real symmetric $n\times n$ matrices, and let $\operatorname{COP}_n$ be the copositive cone. A copositive matrix has 0–1-valued diag…

Let $G$ be a graph, let $M_G$ be its graph matrix, let $F_G(x)=(x^{\circ2})^TM_Gx^{\circ2}$, and let … Define $\vartheta^{(r)}(G)=\min\{t:t(A_G+I)-J\in\mathcal K_n^{(r)}\}$. The fi…