11 problems

Let $f_d(G)$ denote the minimum number of firefighters needed to contain a fire on a vertex-transitive graph $G$ when firefighters may move at most distance $d$ per turn. Square-gr…

Let $d$ be a positive integer and let $\mathbb{Z}^{\Box d}$ denote the $d$-dimensional integer lattice with the local connectivity represented by the Cartesian product. A firefight…

Let $P_n$ be the path graph on $n$ vertices, let $P_n\mathbin{\square}P_n$ be the Cartesian-product square grid, and let $\operatorname{DRMVS}(G,s;d,b)$ denote the maximum number o…

Let $(t_k)$ be the recurrence sequence counting the turns associated with the described two-firefighter strategy, with initial conditions … Recurrence containment conjecture. If th…

Let $f_d(G)$ denote the minimum number of firefighters needed to contain a fire on a vertex-transitive graph $G$ when firefighters may move at most distance $d$ per turn. Hexagonal…

Let the hexagonal grid be the infinite graph whose vertices and edges form the hexagonal grid, and let containment mean that the fire stops spreading after finitely many turns in t…

Consider the continuous firefighter problem in $\mathbb{R}^2$: a path-connected starting fire $F(0)$ contains the origin; at each time increment, the player protects a segment of p…

Hexagonal-grid containability conjecture. The hexagonal grid is not $1$-containable.

Let $P_n\square P_n$ be the Cartesian product of two paths on $n$ vertices, with vertices labelled so that $(0,0)$ is a corner. Let $\mathrm{sn}(G,(0,0))$ denote the maximum number…

Let $P_n$ be the path on $n$ vertices, and let $P_n\square P_n$ be the Cartesian product of two such paths. Write $\rho(G)$ for the expected percentage of vertices saved when a fir…

Let $G_{hex}$ be the infinite hexagonal grid. A firefighting strategy protects one vertex in each time unit, after which the fire spreads to all unprotected neighbors of burning ve…