45 problems

Let a graph be well-quasi-ordered (WQO) under the minor relation when every infinite sequence of graphs contains two graphs such that an earlier one is a minor of a later one. Thom…

Let $(L_i,<_{L_i})$ for $i<\omega$ be an infinite sequence of countable linear orders. A linear order $(L_i,<_{L_i})$ is embeddable into $(L_j,<_{L_j})$ if there is an order-preser…

Daligault–Rao–Thomassé conjecture. If a finitely defined hereditary class of graphs $\mathcal{G}$ is well-quasi-ordered by the induced subgraph relation, then $\mathcal{G}$ has bou…

Let $D_1,D_2,\ldots$ be an infinite sequence of graphic sequences. For graphic sequences $D_1$ and $D_2$, write $D_1\leq D_2$ if there exist graphs $G_1$ and $G_2$ realizing them s…

Let $\Gamma$ be a finite group. A $\Gamma$-gainable quasi-graphic matroid is a quasi-graphic matroid represented by a $\Gamma$-gainable biased framework. An excluded minor is a mat…

For a permutation class $\mathcal{C}$ and an integer $n\geq 1$, say that $\mathcal{C}$ is $n$-well-quasi-ordered (or $n$-wqo) if the set of permutations in $\mathcal{C}$ labeled by…

Let a Robertson chain of length $k$ be the graph obtained from a path of length $k$ by duplicating each edge. A graph contains another graph as a topological minor if the latter ca…

Korpelainen–Lozin–Razgon conjecture. If a hereditary class of graphs $\mathcal{G}$ is defined by a finite set of forbidden induced subgraphs, then $\mathcal{G}$ is well-quasi-order…

Let $\Gamma$ be a finite group and let $\mathcal{G}$ be a minor-closed class of $\Gamma$-gainable biased graphs. An excluded minor for $\mathcal{G}$ is a biased graph outside…

Weak knitwork immersion conjecture. The class $\mathbf{T}(k,d,\Delta,\ell;\Omega)$ is well-quasi-ordered by $\Omega$-knitwork immersion.

Treewidth-bounded immersion conjecture. The class of Eulerian digraphs of treewidth at most $k$ is well-quasi-ordered by immersion.

Weak immersion conjecture. The class of Eulerian digraphs is well-quasi-ordered by weak immersion.

Johnson's conjecture. For every $d\geq 1$, the class of Eulerian digraphs of maximum degree $2d$ is well-quasi-ordered by strong immersion.

Let $k$ be a positive integer, and consider the class of $k$-connected bipartite graphs with the bipartite minor relation. Bipartite-minor non-well-quasi-order conjecture. There ex…

Let $\mathcal{P}_{\mathrm{girth}}$ be the property of a class of matroids that it contains cosimple matroids of arbitrarily large girth. A class of matroids is minor-minimal with r…

Monadic NIP–exponential growth–wqo age conjecture. The following conditions are equivalent: $M$ is monadically NIP; the sequence $\bigl(f_k(\operatorname{Aut}(M))\bigr)$ is bounded…

Let $\mathcal{G}_{{\text{all}}}$ be the class of all graphs, ordered by the minor relation $\leq$. An “$\omega^2$-WQO” is the stronger well-quasi-ordering property referred to in t…

Erdős–Pósa parameter conjecture. For every graph $H$, there exists a minor-monotone graph parameter $\text{\scriptsize\textsf{EP}}_H$ such that $\mathcal{M}_H$ has the Erdős–Pósa p…

The claim concerns countable graphs and the minor relation, in which one graph is obtained from another by deleting vertices or edges and contracting edges. Thomas's conjecture. Th…

Moffatt's conjecture. Ribbon graphs are well-quasi-ordered under the ribbon graph minor relation.

A class of graphs is co-finite if it is the class of graphs excluding a set of finite graphs as minors. A class is UNCOF (Union of Nested Co-finite classes) if there is a sequence…

Let $\mathcal{C}$ be a permutation class. It is strongly algebraic if $\mathcal{C}$ and every subclass of $\mathcal{C}$ have algebraic generating functions, and it is well-quasi-or…

Let $\mathcal{C}$ be a permutation class, meaning a downset under pattern containment. It is well-quasi-ordered if it contains no infinite antichain, and its generating function is…

A hereditary class is a class of finite graphs closed under taking induced subgraphs, and a class is well-quasi-ordered if it has no infinite descending chain or infinite antichain…

A pivot-minor $H$ of a graph $G$ is proper if $|V(H)|<|V(G)|$. A graph $G$ is an intertwine of graphs $H_{1}$ and $H_{2}$ for pivot-minors if it contains both $H_{1}$ and $H_{2}$ a…