2 problems

Let $G$ be a connected chordal graph admitting a tree representation $T$, where $T$ is a subdivided caterpillar. Subdivided-caterpillar Gallai conjecture. Then … This asserts that…

Optimal-size clique transversal conjecture. Every $4$-chordal graph with $n$ vertices has a clique transversal of size at most $2n/7$.