16 problems

Let $G$ be a graph of genus $g$, and let $c(G)$ denote its cop number. Mohar's conjecture. The cop number satisfies … This conjecture proposes the asymptotic growth of cop number i…

Let $G$ be a graph of order $n$, and let $c(G)$ denote its cop number. Weak Meyniel's conjecture. There exists some $\varepsilon>0$ such that … This is explicitly stated to remain…

Andreae–Schroeder conjecture. One has

Genus-square-root conjecture. The cop number of $G$ should be bounded by

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

Let $G$ be a partial cube, and let $c(G)$ denote its cop number. Suppose that $G$ isometrically embeds into the $n$-dimensional hypercube $Q_n$. Partial-cube cop-number conjecture.…

Schröder's conjecture. If $G$ has genus $g$, then

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…

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…

A Steiner triple graph is a tournament obtained from a Steiner triple system on $n$ vertices by cyclically orienting each triple arbitrarily. Nowakowski's conjecture. Every Steiner…

A tournament is an orientation of a complete graph, and $\operatorname{c}(T)$ denotes its cop number. Hill's conjecture. There is an absolute constant $k$ such that, for every tour…

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

Rook graph conjecture. The unique smallest graph for which $c_L=n$ is

The cage conjecture. The $(k,5)$-cage is the unique smallest graph with cop number $k$.