27 problems

Complete saturation and reduction conjecture. Every body $K$ in $\mathbb{E}^d$ (respectively, in $\mathbb{H}^d$) admits a completely saturated packing and a completely reduced cove…

Exact transitive-triple packing conjecture.

Admissibility criterion. A regular spherical polygon is admissible if and only if the inner angle of its polar is at least

Strong-product infinity conjecture. If

Uniqueness conjecture for the $K_{3,3}$ embedding. The proposed embedding of $K_{3,3}$, with coordinates as described in the table, is unique up to an isometry.

Domination-packing conjecture. For every such graph,

Edmonds–Giles conjecture. If the minimum weight of a dicut is $\tau$, then $D$ can pack $\tau$ dijoins.

West–Wu's conjecture. For every positive integer $k$, if $T$ is $3k$-edge-connected in $G$, then $G$ admits $k$ pairwise edge-disjoint $T$-connectors.

Henning–Löwenstein–Rautenbach conjecture. Every connected subcubic graph $G$ except the three graphs $H_1$, $H_2$, and $H_3$ satisfies

Packing version of Dinitz's problem. For $n\geq 3$,

Fractional packing conjectures. There exists a constant $C>0$ such that, for every graph $G$,

NP-hardness conjecture. $C$-$k$-COVER is NP-hard for every convex body $C$ and every integer $k\geq 3$. Similarly, $C$-$k$-PACK is NP-hard for every convex body $C$ and every integ…

Two-arborescence conjecture. If

Kr-free correspondence-cover packing conjecture. For every $r\ge3$, there is some $C_r>0$ such that the following holds for

Triangle-free correspondence packing conjecture.

Let maximal configurations be configurations to which the function $G$ maps permutations, and classify them as efficient or inefficient according to the settlement-planning model.…

Independent packing conjecture. There exists an independent $C$-packing with contact graph $G$.

Let $M$ be a closed $n$-dimensional Riemannian manifold, let $p$ be the exponent in the $p$-Laplacian, and let $c_n(p)$ be the universal constant in the asymptotic law … Here…

Seymour's quarter-integral packing conjecture. Every ideal clutter $\mathcal{C}$ has a $\frac14$-integral packing of value $\tau(\mathcal{C})$.

Gruslys–Leader–Tomon conjecture. The Boolean lattice $2^{[n]}$ can be partitioned into copies of $P$ and a remainder of at most $c=c(P)$ elements.

Let $\beta(\mathbf{K})$ denote the blocking number of a $d$-dimensional convex body $\mathbf{K}$. Zong's blocking-number conjecture. For every $d$-dimensional convex body…

Eventual all-local-maxima junction-occupation conjecture. For a given $G$ and $M(G)$, there is an $N'$ such that for $N\geq N'$, the junction points are always occupied in all fill…

Eventual junction-occupation conjecture. For a given $G$ and $M(G)$, there is an $N'$ such that for $N\geq N'$, the junction points are always occupied in the optimal filling solut…

Nearby-optimal-way conjecture. Given the optimal way of distributing $N-1$ discs, $\{n'_i\}_1^{K}$, the optimal way of distributing $N$ discs is nearby, where nearby means

One-local-maximum-per-way conjecture. There is at most one local maximum per way.