27 problems

Rainbow minimum-threshold conjecture. There exists $n_0$ such that, for every $n\geq n_0$, if

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

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 $K_n$ 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 $G$ be a graph on $n\geq k\geq 2$ vertices, and let $\delta(G)$ denote its minimum degree. The Pósa–Seymour conjecture. If … then $G$ contains the $k$th power of a Hamilton cyc…

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 $C_{n,k,1}$ 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…

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

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…

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 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…

Let $G$ be a $cn$-regular bipartite digraph on $2n$ vertices, meaning that every vertex has indegree and outdegree $cn$. A Hamilton cycle decomposition is a partition of the edge s…

Let $\varepsilon>0$. A graph $G$ has independence number $\alpha(G)$ equal to the maximum size of a set of pairwise nonadjacent vertices, and the square of a Hamilton cycle is the…

Packing conjecture. There exists a constant $C$ such that, if $(k-l)\mid n$ and

Let $k\ge3$ and $1\le d\le k-2$. For an $n$-vertex $k$-uniform hypergraph, let $h_d(k,n)=h_d^{k-1}(k,n)$ be the smallest integer such that minimum $d$-degree at least $h_d(k,n)$ gu…

For $k\geq 2$, let an $(\varepsilon,p,k-1,k)$-pseudorandom graph mean a graph satisfying the stated pseudorandomness condition with parameters $(\varepsilon,p,k-1,k)$. The $k$th po…

Quadratic connectivity conjecture. There exists $C”>0$ such that, for every $k\in\mathbb{N}$, every strongly $C”k^2$-connected tournament contains $k$ edge-disjoint Hamilton cycles…

Quadratic linkedness conjecture. There exists $C'>0$ such that, for every $k\in\mathbb{N}$, every $C'k^2$-linked tournament contains $k$ edge-disjoint Hamilton cycles.

Kühn--Lapinskas--Osthus conjecture. $G$ contains at least

Let $G$ be a graph on $n$ vertices, and let $\operatorname{reg}_{\operatorname{even}}(G)$ denote the largest degree of an even-regular spanning subgraph of $G$. The regular-subgrap…