8 problems

Let $\mathcal F$ be the class of graphs $G$ such that no component of $G$ contains every finite graph as a minor. A graph $U$ is $<$-universal for a class if every graph in the cla…

Let $C$ be a finite or countable graph with no isolated vertices, and decompose it into blocks, its 2-connected components. Let $\tilde C$ be the underlying tree of $C$: its vertic…

Let $C$ be a finite or countable graph with no isolated vertices. Decompose $C$ into its blocks, where each block is a 2-connected component, and let $\tilde C$ be the underlying f…

Let $s(n)$ denote the minimum number of vertices in a universal graph for all trees with $n$ vertices. Asymptotic-limit conjecture. The limit … exists. Determining this limit would…

For $d \in \mathbb{Q}$ and $n \in \mathbb{N}$, let $\mathcal{H}_d(n)$ be the family of all graphs with $n$ vertices and density at most $d$, where the density of a graph $H$ is … A…

Universality conjecture. The following are equivalent:

Quadratic universal graph conjecture. The minimum number of vertices in a bipartite permutation graph containing all $n$-vertex bipartite permutation graphs is

Universal bounded-degree graph conjecture. Let $\Delta\in\mathbb N$ and $p=\omega\left(n^{-1}\log^{1/\Delta}n\right)^{\frac{2}{\Delta+1}}$. Then, with high probability, $G_{n,p}$ i…