21 problems

Let ${\mathcal C}_0$ be a proper minor-closed class of countable graphs. For a minor-closed graph family ${\mathcal C}$, let $\Delta({\mathcal C})$ be the smallest cardinal $k$ suc…

Let $\mathcal{M}$ be a graph class that is proper, minor-closed, and union-closed. For graphs $G$ and $H$, write $G =_{\mathcal{M}} H$ when $\hom(K,G)=\hom(K,H)$ for every…

A graph class $\mathcal{F}$ is proper if some graph is not isomorphic to any graph in $\mathcal{F}$. Write $\int(\mathcal{F})$ for the class of intersection graphs of collections of…

Let $ll in N$ with $ll ge 2$, and let $q$ be the largest prime power with $q le ll$. For sufficiently large $t in N$, let $\falM$ be the class of matroids with no $U_{2,ll+2}$- or…

Let $\mathcal{G}$ be a proper minor-closed graph class, and let $G$ be a connected $n$-vertex graph in $\mathcal{G}$. The spanning tree polytope of $G$ is the convex hull of the in…

For a graph $H$, let $\mathcal{M}_H$ be the class of graphs with no $H$ minor. Write $\raisebox{1.55pt}{\scalebox{1.25}{\chi}}_{\star}(\mathcal{M}_H)$ for its clustered chromatic n…

Frame-template conjecture. There exist $k\in\mathbb Z_+$ and frame templates $\Phi_1,\ldots,\Phi_s,\Psi_1,\ldots,\Psi_t$ such that $\mathcal{M}$ contains each class…

Perturbation conjecture. There exist $k,t\in\mathbb Z_+$ such that each vertically $k$-connected member of $\mathcal M$ is a rank-$(\leq t)$ perturbation of an $\mathbb F$-represen…

Matroid structure theorem. The stated tree-decomposition exists.

Let $t\geq 1$ be an integer. For a graph class $\mathcal{G}$, let $\operatorname{col}_\star(\mathcal{G})$ be the least integer $t$ for which there is a constant $C\geq 1$ such that…

Let $2\leq s\leq t$, let $0\leq p<t$, and write … Here, $K_r$ denotes the complete graph on $r$ vertices, $kK_t$ denotes the disjoint union of $k$ copies of $K_t$, and $G\vee H$ de…

Let $M$ be a matroid. For an integer $k$, $M$ is vertically $k$-connected if, for every $A\subseteq E(M)$ with $\lambda_M(A)<k-1$, either $A$ or $E(M)-A$ is spanning in $M$. Write…

Let $\mathcal{M}$ be a proper minor-closed class of binary matroids. Geelen–Gerards–Whittle's conjecture. There is a polynomial-time algorithm for computing the girth of matroids i…

Geelen–Gerards–Whittle conjecture. Either $\mathcal{G} \subseteq \mathcal{C}$ and

Let $\mathcal{G}$ be an addable, minor-closed class and let $\mathcal{U}\mathcal{G}$ be its corresponding collection of unlabelled graphs. Let $C_n \in_u \mathcal{U}\mathcal{C}_n$…

Let $\mathcal{G}$ be an addable, minor-closed class of graphs, let $G(z)$ be its exponential generating function, and let $\rho$ be the radius of convergence of $G(z)$.…

Growth Rate Conjecture. For every integer $a ge 1$, there is an integer $c>0$ such that at least one of the following holds:

Let $a\geq 1$ be an integer, and let $\mathcal{M}$ be a minor-closed class of matroids. For a matroid $M$, write $r(M)$ for its rank and let $\tau_a(M)$ denote the minimum number o…

Let $q$ be a prime power, and let $\mathcal{M}$ be a base-$q$ exponentially dense minor-closed class of matroids, meaning that its growth rate is eventually of order $q^n$. For a n…

Let $q$ be a prime power, and let $\mathcal{M}$ be a base-$q$ exponentially dense minor-closed class of matroids, meaning that its growth rate is eventually of order $q^n$. Let…

Eventual form conjecture. There exist an integer $k\geq 0$ and an integer $d$ with $$