44 problems

An oriented graph is a directed graph with at most one directed edge between any pair of vertices. For an oriented graph $G$, let $V(G)$ be its vertex set, let $n=|V(G)|$, and writ…

Let $H$ be an $n$-vertex $k$-uniform hypergraph, and let $\delta_{k-1}(H)$ denote its minimum codegree. A tight Hamilton cycle is a Hamilton $\ell$-cycle with $\ell=k-1$, equivalen…

Let $k\ge 3$, let $\delta>1/2$, and let $G$ be a $k$-uniform hypergraph on $n$ vertices with minimum $(k-1)$-degree at least $\delta n$. For $0\le \ell\le k-2$, write…

Let $G$ be a $k$-uniform hypergraph on $n$ vertices with minimum $(k-1)$-degree at least $delta n$ for some $delta>1/2$, and let $Psi(G)$ denote the number of tight Hamilton cycles…

Ai–Guo–Freschi–Lo conjecture. Let $G$ be an oriented graph on $n\geq 3$ vertices. If $\sigma_2(G)\geq n$, then $G$ contains a Hamilton cycle $C$ such that

Let $K_n$ be the complete graph on $n$ vertices, let $C_n$ be a Hamilton cycle on these vertices, and let $C_n^2$ denote its square, obtained by joining pairs at distance at most t…

Let $G_r(n,p)$ be the random $r$-uniform hypergraph, and let $C^{(r)}_{n,\ell}$ be the $r$-uniform $\ell$-cycle on $n$ vertices. Write $Z(C^{(r)}_{n,\ell})$ for the number of copie…

Let $\mathcal{F}$ be the property that $G(n,p)$ contains the square of a Hamilton cycle, meaning a cyclic ordering of the vertices in which every pair of vertices at cyclic distanc…

Let $C_{n,k,1}$ denote a loose Hamilton cycle in an $n$-vertex $k$-uniform hypergraph, where $k-1$ divides $n$. A distribution on embeddings is vertex-spread if it has the vertex-s…

Let $H$ be an $n$-vertex $3$-uniform hypergraph, and let $C_{n,3,2}$ denote its tight Hamilton cycle. A distribution on embeddings is vertex-spread if it has the vertex-spread prop…

For $k\in\mathbb N$, $d\in[k-1]$, $\ell\in\{0,\dots,k-1\}$, and $n$ divisible by $k-\ell$, let $h_{k,\ell,d}(n)$ be the least integer $D\geq0$ such that every $n$-vertex $k$-unifor…

For a positive real number $\alpha$ and $k\in\mathbb N$, let $\delta^{\mathrm{CON}}_{k,\ell,d}$ be the threshold appearing in the main vertex-spread theorem, let…

Let $\delta^{\mathrm{CON}}_{k,\ell,d}$ denote the minimum $d$-degree threshold used in the preceding result, and let $\delta_{\mathcal{C}_{\ell,k},d}$ be the corresponding minimum…

Strong rainbow connectivity conjecture. If $n$ is sufficiently large and $\mathbf{T}$ is strongly rainbow-connected, then $\mathbf{T}$ contains a transversal Hamilton cycle.

Connecting-barrier conjecture. Given $\mu>0$ and $r\geq 4$, there exists $\alpha>0$ such that the following holds for sufficiently large $n$. If

Staden–Treglown conjecture. For every $\mu>0$, there exist $\alpha>0$ and $0in mathbb{N}$ such that the following holds. For every $n$-vertex graph $G$ with $geq n0$, if

Loose-Hamilton-cycle threshold conjecture. For every $\gamma>0$ and every $\Delta\in\mathbb N$, there exists $n_0\in\mathbb N$ such that every $k$-graph $H$ on $n\geq n_0$ vertices…

Equality conjecture. For all $k>j\geqslant 1$,

Bailey–Stevens conjecture. The complete hypergraph $K_n^k$ has a decomposition into tight Hamilton cycles if and only if

Rainbow tight Hamilton cycle conjecture. For every $k\geq 3$ and $\gamma>0$ there exist $c>0$ and $n_0>0$ such that if $(\mathcal{H},\phi)$ is an $n$-vertex colored $k$-graph with…

Daykin's conjecture. There is a positive constant $bc$ such that, for every $n$, every locally $bc n$-bounded coloring of $K_n$ contains a properly colored Hamilton cycle.

Fix an integer $k\geq 2$ and a constant $\gamma>0$. Let $G$ be an $n$-vertex $k$-graph with minimum codegree … A tight Hamilton cycle is a cyclic ordering of the vertices of $G$ su…

Let $G$ be an $n$-vertex graph with minimum degree at least $n/2$, so that $G$ is a Dirac graph. A Hamilton cycle is a cycle containing every vertex of $G$ exactly once. Minimum Ha…

Let $G$ be a $cn$-regular tripartite digraph with three vertex classes, each of size $n$, meaning that every vertex has indegree and outdegree $cn$. A Hamilton cycle is a directed…

Let $G$ be a diregular bipartite tournament on $2n$ vertices, so $G$ is a complete bipartite orientation with equal indegree and outdegree at every vertex. Edge-disjoint Hamilton c…