4 problems

For integers $t\geqslant 2$ and $k\geqslant 2$, define … Here, $\operatorname{td}(G)$ is the treedepth of $G$, $\operatorname{tdtwo}(G)$ is its 2-treedepth, and $P_t$ denotes the p…

Let $r\geq 4$. A graph is $C_{2r}$-subgraph-free if it contains no subgraph isomorphic to the cycle $C_{2r}$. The diameter of a graph is the maximum distance between two vertices.…

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…

Let $G$ be a graph, let $p$ be its pathwidth, and let $\ell$ be a positive integer. A path of order $2^\ell$ is a path with $2^\ell$ vertices. Pathwidth–treedepth conjecture. Every…