Obstructions for bounded shrub-depth and rank-depth

Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph. It is well known that a graph has large tree-depth if and only if it has a long path as a subgraph. We prove an analogous statement for shrub-depth and rank-depth, which was conjectured by Hlin\v{e}n\'y, Kwon, Obdr\v{z}\'alek, and Ordyniak [Tree-depth and vertex-minors, European J.~Combin. 2016]. Namely, we prove that a graph has large rank-depth if and only if it has a vertex-minor isomorphic to a long path. This implies that for every integer $t$, the class of graphs with no vertex-minor isomorphic to the path on $t$ vertices has bounded shrub-depth.Comment: 19 pages, 5 figures; accepted to Journal of Combinatorial Theory Ser.

Local properties of graphs with large chromatic number

This thesis deals with problems concerning the local properties of graphs with large chromatic number in hereditary classes of graphs. We construct intersection graphs of axis-aligned boxes and of lines in $\mathbb{R}^3$ that have arbitrarily large girth and chromatic number. We also prove that the maximum chromatic number of a circle graph with clique number at most $\omega$ is equal to $\Theta(\omega \log \omega)$. Lastly, extending the $\chi$-boundedness of circle graphs, we prove a conjecture of Geelen that every proper vertex-minor-closed class of graphs is $\chi$-bounded