22 problems

Mattman–Pierce conjecture. The obstruction set $\operatorname{obs}(\mathcal{C}_k^{\mathcal{K}})$ contains the $Y\Delta Y$-families of $K_{n+5}$ and $K_{3^2,2^n}$.

For an integer $ell\geq 3$, let $\mathscr{F}_{\ell}$ be the family of cycles of length at least $ell$ in a graph $G$. A set $X\subseteq V(G)$ is a transversal of…

Thomas's conjecture. For every graph $H$, the family of $H$-minors has the half-integral Erdős–Pósa property.

Ahn–Gollin–Huynh–Kwon conjecture. The upper bound can be improved to

Let $H$ be a planar graph. An $H$ minor-model is a model of $H$ in $G$, and minor-models are pairwise distance at least $d$ when every two distinct models are at that distance or f…

Let $H$ be a planar graph. A $K$-fat minor-model of $H$ is a $K$-fat model of $H$, and $B_G(Z,cK)$ denotes the ball of radius $cK$ around $Z$ in $G$. Then there exists a constant…

Let $G$ be a graph, and let $d$ be the prescribed distance separating cycles. Let $f(k)$ be the function in bounding the size of a vertex set meeting every collection of $k$ cycles…

Infinite-group characterization conjecture. The family $\mathcal{F}$ satisfies the half-integral Erdős–Pósa property, and it satisfies the Erdős–Pósa property if and only if…

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…

For a graph $H$, $kH$ denotes the disjoint union of $k$ copies of $H$, and a graph is $H$-induced-minor-free if it does not contain $H$ as an induced minor. The planar induced-mino…

Let $G$ be a graph and $S\subseteq V(G)$. An $S$-cycle is a cycle containing a vertex of $S$, and an induced packing is a collection of cycles with no edge between distinct cycles.…

For a graph $G$, an induced packing of cycles is a collection of cycles with no edge between distinct cycles. For a vertex set $X$, let $B_G(X,1)$ be its closed distance-one neighb…

For a positive integer $d$, a distance-$d$ packing of cycles in a graph $G$ is a set of cycles such that no path of length at most $d$ joins two distinct cycles. For a vertex set…

Let $G$ be a graph, let $X,Y\subseteq V(G)$, and for a vertex set $S$ and integer $d$ let $B_G(S,d)$ be the vertices at distance at most $d$ from $S$. An $(X,Y)$-path is a path wit…

Bounded-genus universal-obstruction conjecture. One has

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…

Huynh–Joos–Wollan conjecture. Even for $k>2$, the family of $(\Gamma_1,\Gamma_2,\ldots,\Gamma_k)$-non-zero cycles has the half-integral Erdős–Pósa property.

Condensed-wall conjecture. If there is an integer $r$ such that the condensed wall of size $r$ contains an $H$-expansion, then the family of $H$-expansions has the edge-Erdős–Pósa…

Large-expansion conjecture. There is an integer $c$ such that for every planar graph $H$ of treewidth (or even pathwidth) at least $c$, the family of $H$-expansions does not have t…

The $3k-3$ bound conjecture. If $G$ does not contain $k$ pairwise vertex-disjoint even $A$-paths, then there is a set of at most $3k-3$ vertices that meets every even $A$-path.

Long $S$-cycle Erdős–Pósa conjecture. For every graph $G$, every subset of vertices $S$, and every pair of positive integers $k,\ell$, there is either a set of $k$ disjoint $S$-cyc…

Let $\ell\in\mathbb{N}$ be fixed, and call a cycle long if its length is at least $\ell$ and short otherwise. The lemma preceding this conjecture gives a vertex set of size governe…