6 problems

Schröder's conjecture. The cop number of a genus-$g$ graph satisfies

Conjecture. If $G$ is $P_k$-free for some $k\ge 4$, then $ccn(G)\le k-3$.

Conjecture. For every connected planar graph $G$ on at most 19 vertices, one has $ccn(G)\le 2$.

Conjecture. For every connected $P_5$-free graph $G$, one has $ccn(G)\le 2$.

Let $D$ be the planar graph constructed by Gao and Yang, and let $cop_{1}(D)$ denote its 1-cop-move number. Gao–Yang conjecture. The graph $D$ satisfies … Gao and Yang had construc…

Let $G$ be a finite connected planar graph with $n$ vertices, and let $cop_{1}(G)$ denote the minimum number of cops needed to catch a robber when at most one cop may move by one s…