3 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…

Let $\mathbf{G}=\{G_1,\dots,G_{2s-1}\}$ be a collection of matchings, each of size $s$. A matching partial $\mathbf{G}$-transversal of size $s$ is a matching with $s$ edges whose e…

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