15 problems

Scott's conjecture. For every fixed graph $H$, the class of graphs containing no induced subdivision of $H$ is chi-bounded.

A graph is ISK4-free if it contains no induced subdivision of $K_4$, and it is triangle-free if it contains no induced subgraph isomorphic to a triangle. 3-color conjecture for tri…

Here graphs are simple and finite. A graph is triangle-free if it contains no triangle and ISK4-free if it contains no induced subdivision of $K_4$, where $K_4$ is the complete gra…

All graphs under consideration are finite and simple. A graph is ISK4-free if it contains no induced subdivision of $K_4$. Lévêque et al.'s 4-color conjecture. Every ISK4-free grap…

For a graph $H$, an odd subdivision is a subdivision in which every replacing path has an odd number of edges. Scott–Seymour's conjecture. For every graph $H$ and integer $t$, the…

For a graph $H$, let $\mathcal{F}_H$ be the class of graphs with no induced subdivision of $H$. A class is polynomially degree-bounded if there is a polynomial $p$ such that every…

For graphs $G$ and $H$, say that $G$ is $H$-subgraph-free if it does not contain $H$ as a subgraph, and say that a graph is an induced subdivision of $H$ if it is obtained from $H$…

For positive integers $\ell$ and $t$, a $t$-theta graph is a graph consisting of two vertices joined by $t$ internally vertex-disjoint paths. A long balanced $t$-theta is an induce…

Let $H$ be a graph, and for each integer $s$ let $p(s,H)$ be such that every graph without a copy of $K_{s,s}$ and with average degree at least $p(s,H)$ contains an induced subdivi…

A multigraph $H$ is widespread if, for every subdivision $H'$ of $H$ and every $κ e0$, there exists $r$ such that every $r$-bounded ideal of $H'$-subdivision-free graphs with cliqu…

A multigraph is pervasive if, in the relevant ideal setting, excluding every subdivision of it as an induced subgraph yields a $χ$-bounded ideal. A chandelier is obtained from a tr…

A graph $H$ is weakly pervasive if the ideal of graphs containing no subdivision of $H$ as an induced subgraph is $χ$-bounded. Scott's conjecture. Every graph is weakly pervasive.…

All graphs in this paper are finite and simple. For a graph $H$, a graph $G$ contains $H$ if $H$ is isomorphic to an induced subgraph of $G$, and $G$ is $H$-free otherwise. An indu…

Monotonicity conjecture. If $i_D$ (respectively, $i'_D$) is NP-complete, then for every digraph $D'$ containing $D$ as an induced subdigraph, $i_{D'}$ (respectively, $i'_{D'}$) is…

Induced-subdivision dichotomy conjecture. $i_D$ is NP-complete unless $D$ is the disjoint union of spiders and at most one $2$-cycle.