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

A graph is $5$-vertex-critical if it has chromatic number $5$ and deleting any vertex lowers its chromatic number; a graph is co-gem-free if it has no induced subgraph isomorphic to the complement of the gem. The paper reports a list of 327 such graphs.

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

The finiteness of $k$-vertex-critical co-gem-free graphs is stated to remain open for all $k\geq 5$. The list was generated computationally for graphs of order at most 18, with no examples of order 13 through 18.