3 problems

Let $P_7$ denote the path on seven vertices and let $C_3$ denote the cycle on three vertices. A graph is 4-vertex-critical if its chromatic number is $4$ and deleting any vertex lo…

Finite-list conjecture. There are only finitely many $5$-vertex-critical co-gem-free graphs, and the list of 327 is complete.

Dichotomy conjecture. There is a finite number of $k$-vertex-critical $H$-free graphs if and only if $H$ is an induced subgraph of $P_4+sP_1$ for some $s\geq 0$.