13 problems

Let $G$ be a graph, with vertex-connectivity $\kappa(G)$ and independence number $\alpha(G)$. A graph is pancyclic if it contains cycles of every length from $3$ to $|V(G)|$. Jacks…

Let $G$ be a Hamiltonian graph, let $\alpha(G)$ denote its independence number, and let $n$ be its number of vertices. Erdős's conjecture. Every Hamiltonian graph with…

A 4-connected planar graph is a planar graph with at least five vertices and no vertex-cut of size less than $4$. A graph is pancyclic if it has a cycle of every length from $3$ th…

Hilton's conjecture. For every integer $\ell$ with $3\le \ell\le n$,

Let $p$ be an integer with $3\leq p\leq 5$. A graph $G$ is an $[s,t]$-graph if every induced subgraph of $G$ of order $s$ has size at least $t$. A graph is pancyclic if it contains…

A graph is weakly pancyclic if the set of lengths of its cycles forms an interval. Brandt's conjecture. Every non-bipartite graph on $n$ vertices with more than … edges is weakly p…

A digraph $D$ has order $n\ge 3$. A pair of nonadjacent vertices is dominated if it has a common in-neighbor, and dominating if it has a common out-neighbor. Let $d(u)$ denote the…

Fujita and Magnant's conjecture. If

Erdős's conjecture. There are constants $c_1$ and $c_2$ such that, for all $k$,

Conjecture 1.9. $D$ is pancyclic, meaning that it contains directed cycles of every length from $3$ to $n$.

Manoussakis's conjecture. If, for every triple $x,y,z\in V(D)$ with $x$ nonadjacent to $y$, the inequalities

Let $n\geq 3$, let $k$ be an integer, and let $G$ be a graph of order $n$ with minimum degree … A $k$-edge colouring is a decomposition … where each $G_i$ is a spanning colour clas…

Amar–Fournier–Germa's triangle-free conjecture. If $G$ is triangle-free, then $G$ contains cycles of length $\ell$ for every $4 \leq \ell \leq n$, with the same conclusion as in th…