4 problems

Continuous standard repair optimality conjecture. The factor

The UMR conjecture for $H_{n,m}$. Among all $m$-cyclic graphs, $H_{n,m}$ is UMR with respect to node cop-win reliability for all $n$ sufficiently large.

Kochar–Xu conjecture. If $(\lambda_1,\ldots,\lambda_n)\prec(\theta_1,\ldots,\theta_n)$, then

Let $K_n$ be the complete graph on $n$ vertices, and let $R_{K_n}(x)$ denote its reliability polynomial. Write ${\cal D}=\{z:|z-1|\leq 1\}$. Sokal's conjecture. The polynomial…