113 problems

Feder–Subi conjecture. For every $d \in \mathbb{N}$, every $2$-edge-colouring of $Q_d$ contains vertices $u,v$ with $\operatorname{dist}(u,v)=d$ that are connected by a path with a…

Let $Q_n$ be the $n$-dimensional hypercube, and let $W(Q_n)$ denote the maximum number of colors in an interval coloring of $Q_n$. Petrosyan's conjecture. … for any…

Norine's conjecture. Every antipodal $2$-edge-colouring of $Q_n$ contains a monochromatic path from some vertex to its antipode.

Let $Q_n$ be the $n$-dimensional hypercube, with $n\geq 2$, and let a perfect matching be a matching that covers every vertex of $Q_n$ exactly once. Kreweras's conjecture. Any perf…

Let $Q_n$ be the graph with vertex set $\{0,1\}^n$ and edges between vertices differing in exactly one coordinate. A path decomposition of $Q_n$ is a pairwise edge-disjoint union o…

Let $\vv{Q_d}$ denote the oriented $d$-dimensional hypercube, and let $\vv{r}(\vv{Q_d})$ be its oriented Ramsey number. Directed Burr–Erdős conjecture. There is an absolute constan…

Let $Q_d$ be the $d$-dimensional hypercube, and let $varphi$ be a partial $d$-edge coloring of $Q_d$. A color class is the set of edges receiving one fixed color, and an induced ma…

Let $FQ_n$ be the $n$-dimensional folded cube, let $E^m$ be a subset of $2^n-1$ edges of $FQ_n$, and let $FQ_n-E^m$ denote the graph obtained by deleting those edges. A perfect mat…

For $k\in\mathbb{N}$, let $Q_{2k+1}$ denote the $(2k+1)$-dimensional hypercube. A graph is path-pairable if every pairing of its vertices can be joined by pairwise edge-disjoint pa…

Let $Q_n$ denote the $n$-dimensional hypercube, let $Q_n^p$ denote its $p^{\textrm{th}}$ graph power, and let $D(G)$ be the distinguishing number of a graph $G$, namely the smalles…

Let $Q_n$ be the $n$-dimensional hypercube and let $F_{n+1}$ be the dual-cube of dimension $n+1$. Let $k$ be a positive integer. Dual-cube lifting conjecture. If $Q_n$ has $k$ comp…

Let $[k]^d$ be the $d$-dimensional grid, with $k\in\mathbb{N}_{\geq 2}$ and $d\in\mathbb{N}$, and let $D(k,[k]^d)$ denote the maximum density of a configuration avoiding the releva…

Norine's and related conjectures. For every $n$, the following assertions hold:

Let $Q_n$ be the $n$-dimensional hypercube, let $C_4$ be the cycle on four vertices, and let $ex(Q_n,C_4)$ denote the maximum number of edges in a $C_4$-free subgraph of $Q_n$. Erd…

Let $Q_d$ be the $d$-dimensional hypercube, and let $m(Q_d;4)$ denote the minimum cardinality of a set that percolates under the $4$-neighbour bootstrap process on $Q_d$. Exact-siz…

Let $Q_7$ be the 7-dimensional hypercube and let $C_4$ denote the cycle of length four. Write $\mathrm{ex}(Q_7,C_4)$ for the maximum number of edges in a subgraph of $Q_7$ containi…

Let $Q_8$ be the 8-dimensional hypercube and let $C_4$ denote the cycle of length four. Write $\mathrm{ex}(Q_8,C_4)$ for the maximum number of edges in a subgraph of $Q_8$ containi…

Casselgren, Markström and Pham's conjecture. If $n$ and $d$ are positive integers, and $\varphi$ is a proper edge-precoloring of $K_{n,n}^d$ with at most $nd-1$ precolored edges, t…

Let $\mathcal P$ be a finite poset. Its hypercube-width $w^*(\mathcal P)$ is the minimum $w^*\in\mathbb{N}$ such that there exists an induced copy of $\mathcal P$ in…

Casselgren–Markström–Pham conjecture. Any precolored distance-2 matching in $Q_d$ can be extended to a proper $d$-edge coloring.

Relative rainbow Turán conjecture for hypercubes. The hypercube has the property that

Brimkov's conjecture. For every integer $d\geq 2$,

Hung's conjecture. An $n$-dimensional augmented cube admits a Hamiltonian decomposition.

For a graph $G=(V,E)$, a multipacking is a set $M\subseteq V$ such that, for every vertex $v\in V$ and every integer $k\geq 1$, $|N_k[v]\cap M|\leq k$, where $N_k[v]$ is the set of…

Asymptotic hypercube burning conjecture. If $k$ is a positive integer, then there exists an integer $n_k$ such that