2 problems

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The finite-list conjecture for 5-vertex-critical co-gem-free graphs
Finite-list conjecture. There are only finitely many $5$ -vertex-critical co-gem-free graphs, and the list of 327 is complete.
Conjectureopenforbidden-subgraphs graph-theory vertex-critical-graphsarXiv posted 2 days ago
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Dichotomy conjecture for finite families of vertex-critical H-free graphs
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 $s\geq 0$ .
Conjectureopenforbidden-induced-subgraphs graph-theory vertex-critical-graphsarXiv posted 2 days ago

The finite-list conjecture for 5-vertex-critical co-gem-free graphs