39 problems

Hasunuma's conjecture. If

For integers $n$ and $c$, let $f(n,c)$ be the least integer $d$ such that every $n$-vertex graph with minimum degree at least $d$ contains a cycle with at least $c$ chords. Kára an…

For a digraph $D$, its minimum out-degree is the minimum number of outgoing edges over all vertices of $D$. Lichiardopol's conjecture. For every $k\geq1$, there exists an integer…

Odd-cycle decomposition threshold conjecture.

All graphs under consideration are finite and simple. For a graph $G$ and a vertex $v$, let $d_G(v)$ denote the degree of $v$. Dean's conjecture. For every integer $k \ge 3$, every…

Let $W_{2k}$ be the wheel on $2k+1$ vertices, let $G$ be a graph on $n$ vertices, and let $\delta(G)$ denote its minimum degree. Minimum-degree wheel conjecture. For all integers…

El-Zahar's conjecture. If

For $r\geq2$ and $\delta\in(0,1)$, let … where $\mathcal{G}(n,r,\delta)$ consists of the $n$-vertex $r$-edge-coloured graphs with $\delta(G)\geq(1-\delta)n$, and $tc(G)$ is the sma…

For $r\geq2$ and $\delta\in(0,1/2)$, let … where $\mathcal{G}(n,r,\delta)$ consists of the $n$-vertex $r$-edge-coloured graphs with $\delta(G)\geq(1-\delta)n$, and $cp(G)$ is the s…

Let $G$ be an $n$-vertex $r$-edge-coloured graph, and let $tc(G)$ be the smallest number of not necessarily vertex-disjoint monochromatic trees whose vertices cover $V(G)$. Bal–DeB…

Let $G$ be a $k$-graph on $n$ vertices. Write $\delta_1(G)$ for its minimum vertex degree, the minimum number of edges containing any one vertex. Spanning-component conjecture. If…

Stein's conjecture. If

Aragão–Marciano–Mendonça's conjecture. If

Favaron–Shi's conjecture. If $G$ is a minimal $k$-factor-critical graph, then

Let $G$ be a graph on $n$ vertices, let $\delta(G)$ denote its minimum degree, and let a cycle of maximum order mean a cycle containing the maximum possible number of vertices in…

Let $X$ and $Y$ be connected graphs on $n$ vertices. For a graph $G$, write $\delta(G)$ for its minimum degree, and let $\operatorname{FS}(X,Y)$ denote the friends-and-strangers gr…

Liu–Ma conjecture. $G$ contains $eil k/2$ cycles with consecutive odd lengths.

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…

Jung's reverse minimum-degree conjecture. If $\overline{p}\geq\Greekmath 0114 -1$, then

Minimum-degree construction-time conjecture. The conclusion of Theorem holds for $k=o(n)$.

Let $H$ be an ordered graph. An $H$-tiling in an ordered graph $G$ is a collection of vertex-disjoint copies of $H$ in $G$, and let $\delta(G)$ denote the minimum degree of $G$. Wr…

Average minimum-degree conjecture. For every $k\geq3$ and $\varepsilon>0$, there exists $n_0=n_0(k,\varepsilon)$ such that for every $n\geq n_0$, if there are numbers…