12 problems

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…

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…

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…

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…

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…

Thomas's conjecture. For every graph $H$, there exists a function $f$ such that for every graph $G$ and every positive integer $k$, either $G$ half-integrally packs $k$ graphs each…

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…