22 problems

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…

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…

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…

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…

Let $\mathcal{C}$ be a permutation class, and let $\langle\mathcal{C}\rangle$ denote its substitution closure. The substitution-closure conjecture. If $\langle\mathcal{C}\rangle$ i…

Let $\mathcal{C}$ be a permutation class, and let $\mathcal{C}^{+1}$ denote its one-point extension. The 2-wqo one-point-extension conjecture. If $\mathcal{C}$ is 2-wqo, then…

Let $\mathcal{C}$ be a permutation class, and let $\mathcal{C}^{+1}$ denote its one-point extension. A class is lwqo when it is well-quasi-ordered under labeling by arbitrary wqo l…

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 $k$ be a positive integer. Let $G_1,G_2,\ldots$ be graphs that do not contain a Robertson chain of length at least $k$ as a topological minor. Let $Q$ be a set equipped with a…

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…

Let $H$ be a finite group. A gain-graphic matroid is a frame matroid obtained from a graph with edge gains in $H$, and a class is well-quasi-ordered when it has neither an infinite…

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…

Korpelainen–Lozin–Razgon conjecture. A class of graphs which is wqo by the induced subgraph relation is lwqo if and only if it is defined by finitely many minimal forbidden induced…

Strong rationality conjecture. A permutation class is strongly rational if and only if it is well-quasi-ordered and does not contain the class of $312$-avoiding permutations or any…

A hereditary graph class with finite distinguishing number is one whose distinguishing number, as defined in the source, is finite. The induced subgraph relation is the ordering by…

Let $\mathcal{C}$ be a hereditary graph class, meaning that it is closed under taking induced subgraphs, and suppose that $\mathcal{C}$ is well-quasi-ordered under the induced subg…

Let a graphic degree sequence be a multiset of non-negative integers realized as the degree sequence of a simple graph. For a degree sequence $D$, let … be its set of realizations.…

Well-quasi-ordering conjecture. Any infinite set of $\mathrm{GF}(q)$-representable matroids contains two matroids, one of which is isomorphic to a minor of the other.