67 problems

Let a non-parallel cylinder packing be a packing of congruent infinite circular cylinders in $\mathbb{R}^3$ such that no two cylinder axes are parallel. For a packing $\mathcal C$,…

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.

Let $C$ be a centrally symmetric convex body in $E^n$. A family of copies of $C$ is pairwise touching if every two members touch, and copies are homothetic when each is obtained fr…

Lattice-packing conjecture. The densest packings of the centrally symmetric Platonic and Archimedean solids are given by their corresponding optimal lattice packings.

Two-saturated disk-packing density conjecture.

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.

Let $G$ be a planar graph. Write $\rho(G)$ for its packing number and $\gamma(G)$ for its domination number. Ducz–Gujgiczer's conjecture. The bound … is optimal for planar graphs.…

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

Let $f(n)$ denote the maximum sum of the side lengths of $n$ non-overlapping squares packed inside a unit square. Erdős's square packing conjecture. For every positive integer $n$,…

Let $k$ be a positive integer, and let $k^{2}+1$ non-overlapping squares lie inside a unit square. The circumference of a square is its perimeter, and the total circumference is th…

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,

Henning et al.'s conjecture.

Shepherd–Vetta conjecture. The function $f(\tau)$ is of order $\Omega(\tau)$.

Three-dijoin threshold conjecture. There exists an integer $\tau\geq 3$ such that every digraph with minimum dicut size at least $\tau$ contains $3$ disjoint dijoins.

Let $F=(V,E\cup A)$ be a mixed graph, let $k\in\mathbb Z_{>0}$, and let $\ell:K\rightarrow\mathbb Z_{>0}$. For a subpartition $\mathcal{P}$ of $V$, write…

Let $\mathcal C=\{C_i\}_{i=1}^{\infty}$ be a non-parallel cylinder packing in $\mathbb{R}^3$, and let $(\delta^{*})^{-}(\mathcal C)$ denote its translation-invariant lower density.…

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

Soifer's conjecture. For every triangle $T$,