11 problems

Full stability classification conjecture. For $1<p\leqslant 5$, the complete conclusion of the cubic Theorem should hold verbatim, with the same case-differentiation inequalities i…

Let $G$ be the tensor product of two distinct star graphs $S_n$ and $S_m$. Star-graph tensor-product conjecture. The $b$-chromatic number of $G$ is $m+n+1$. This conjecture propose…

Let $K_{1,n}$ be the star on $n+1$ vertices, and let $r(K_{1,n};k)$ and $r_\ell(K_{1,n};k)$ denote its $k$-color Ramsey number and list Ramsey number, respectively. Alon–Bucić–Kalv…

Let $S_n$ be the star graph on $n$ vertices, with central vertex labeled by $1$, and let … For a vertex subset $S$, write $Y_S$ for the associated cluster-algebra element and…

Let $S_{2n}$ be the star graph with $2n$ leaves, let $p$ be the initial density parameter, and let $T_p^2(S_{2n})$ denote the annihilation time for the two-type system on this grap…

Consider a star graph with one incoming edge and $N-1$ outgoing edges, and a shifted state whose central peak is on the incoming edge. Perturbations that break the symmetry between…

Let a star graph have an even number of edges, and consider the shifted stationary states of the nonlinear Schrödinger equation, with the action functional defined by the variation…

Excited-state instability conjecture. Every excited state for $N\geq3$ and $\omega\in\mathbb{R}$ is unstable in $W(\Gamma)$.

Local-minimum conjecture. When $N\geq3$ and $0<\gamma\leq\gamma^{\ast}(N)$, the action $S_{\omega,\gamma}$ has a local constrained minimum that is larger than the infimum.

Local constrained-minimum conjecture. When this condition on $\alpha$ fails, the action has a local constrained minimum whose value is larger than

Let $\Gamma$ be an $N$-armed star graph with sector angles $\beta_1,\ldots,\beta_N$ satisfying the geometric constraints for a star, and let the ground-state energy of the correspo…