7 problems

Bollobás–Meir conjecture. For any finite set of points $X\subseteq[0,1]^k$, there exists a Hamiltonian cycle $H$ on $X$ with $S_k(H)\leq 2\cdot k^{k/2}$ if $k\neq3$, and…

Let $P_{\text{ASEP}}^{n}$ be the asymmetric subtour elimination polytope, let $\boldsymbol{x}$ be a vertex of this polytope, and let $\mathcal{N}(\boldsymbol{x})$ denote its neighb…

Bertsimas–Grigni conjecture. For every linear order $\leq$ on the unit square,

Hamiltonian-path conjecture. The equalities

Let an instance have a fixed number of vertices, and let $x^*$ be an optimal fractional solution of its subtour LP. The two-disjoint-triangles conjecture. The instances maximizing…

Consider extreme points of the Held–Karp relaxation on $n$ vertices, and let the maximum degree of an extreme point mean the maximum degree of its support graph. The maximum-degree…

Let $P_n$ be the Graphical Traveling Salesman Polyhedron, let $\mathcal G_B$ be the graph associated with a relaxation $\mathcal R_B$, and call a facet of $P_n$ an NR-facet when it h…