4 problems

Let $G$ be a cocomparability graph, meaning that its complement admits an acyclic transitive orientation. Define $LexCycle(G)$ as the maximum length of a cycle of vertex orderings…

Conjecture. $G$ is an interval $k$-graph if and only if it has no induced subgraph isomorphic to $\overline{C}_6$ or $\overline{2P}_3$.

Dusart–Habib conjecture. A similar approach should recognize cocomparability graphs using only a constant number of sweeps.

Golumbic–Monma–Trotter conjecture. The intersection of cocomparability graphs with tolerance graphs is exactly the class of bounded tolerance graphs.