7 problems

Let $\mathbb Z^2_{\triangle}$ denote the triangular lattice, and let $q_{\mathbb Z^2_{\triangle}}$ be its containment threshold in the firefighter problem. The triangular-grid fire…

Consider the Pyro game on the infinite Cartesian grid, with the original burned vertex at $(0,0)$ and one firefighter protecting vertices over successive steps. Here, a vertex's di…

Let the infinite Cartesian grid be the graph whose vertices are the integer lattice points and whose edges join lattice points at grid distance one. In the firefighter model, firef…

Let $G$ be a planar graph. Say that a graph is $2$-flammable if the fire can be controlled under the firefighter process with two firefighters available at each step. The planar no…

Let $G$ be a planar graph, and let $\rho_3(G)$ denote its surviving rate when three firefighters are available at each step. The planar surviving-rate conjecture. There exists…

Let $\mathcal{P}_5$ be the class of planar graphs of girth at least $5$, and let $\mathit{ff}(\mathcal{P}_5)$ denote its firefighter number. Firefighter number conjecture for plana…

Let $\mathcal{P}$ be the class of planar graphs, and let $\mathit{ff}(\mathcal{P})$ denote its firefighter number. Firefighter number conjecture for planar graphs. … The paper note…