10 problems

Minimal-order conjecture for the attacking-cop-number gap. If

Strong attacking-cop-number conjecture. For all integers $k \geq 4$ there exists a graph $H$ such that $\operatorname{c}(H)=k$ and

Turcotte's conjecture. For any positive integer $p$ and any graph $G$ such that $\alpha(G)<p$, we have

Let $S$ be a geodesic surface of genus $g$, and let $c(S)$ denote its cop number. Mohar's surface conjecture. … The paper describes this as a tough conjecture because it implies th…

Let $G$ be a graph, and write $c(G)$ for its cop number. The graph $G$ is $2P_2$-free if it has no induced subgraph consisting of two vertex-disjoint copies of the path on two vert…

Andreae–Schroeder conjecture. One has

Let $G$ be an undirected cop-win graph, and consider optimal play, meaning that the cop minimizes capture time while the robber maximizes it. Hahn's conjecture. The distance betwee…

An oriented graph $D$ is a directed graph obtained by orienting the edges of a simple undirected graph; a directed cycle is cop-dominated if the robber cannot avoid capture by ente…

Ten-vertex classification conjecture. The only graphs $G$ on 10 vertices with $c_L(G)=3$ are either the Petersen graph, or a copy of $R_3$ with an additional vertex $v$ whose degre…

Maximum-degree conjecture. If