13 problems

Strong conjecture on $F$-irregular graphs. For each connected graph $F$ of order $|F| \geq 3$, there exist infinitely many $F$-irregular graphs.

Chartrand–Holbert–Oellermann–Swart conjecture. For every connected graph $F$ of order at least $3$, there exists a nontrivial $F$-irregular graph.

Spectral bowtie conjecture. For $q\ge 3$, if $G$ is a graph of sufficiently large order $n$ satisfying

For each positive integer $k$, let $c(k)$ denote the exponent for counting copies of an arbitrary $k$-vertex tournament. Recursive counting conjecture. For all $k \ge 3$ it holds t…

For a positive integer $k$, let $c(k)$ be the exponent for counting copies of an arbitrary $k$-vertex tournament, and let $c^*(k)$ be the exponent for counting copies of the transi…

For a positive integer $k$, let $c(k)$ be the exponent for counting copies of an arbitrary $k$-vertex tournament, and let $c^*(k)$ be the exponent for counting copies of the transi…

Polynomial homomorphism-growth conjecture. For some integer $k=k(H)$,

Asymptotic quasi-complete bipartite extremal conjecture. For any integers $e,n$ with $\omega(n)=e\le \frac{n^2}{4}$ and any bipartite graph $H$, we have

Let $\Phi$ be a uniformly monotone property, and let $\min(\Phi)$ denote its class of minimal graphs. Assume that there is no constant $c$ such that every graph in $\min(\Phi)$ has…

Let $\mathcal{C}$ be a class of graphs with unbounded treewidth, and let $\textup{\textbf{p-}}\textsc{Sub}(\mathcal{C})$ denote the parameterised subgraph decision problem for the…

Let $\Phi$ be a uniformly monotone property, and let $\min(\Phi)$ denote its class of minimal graphs. Vertex-cover number measures the minimum size of a vertex set meeting every ed…

Uniqueness conjecture. For every $K\geq4$, there is only one subgraph-counting identity modulo the easily described families and the difficult identities for lower $K$.

Existence conjecture. For every $K\geq 4$, there is such an identity that holds for every graph $G$ with $n$ vertices, has one graph $J_{a,i}$ or $J'_{a,j}$ with $K$ vertices, and…