5 problems

Finite-presentation conjecture. $G$ has either of the following two equivalent finite presentations:

Stone-parity conjecture. The sign of $\partial R$ corresponds to the parity of the minimum number of stones needed for $R$ to be signed tilable when adding tiles of either weight a…

Let the hexagonal grid be the infinite 3-regular graph whose vertices and edges arise from a tiling of the plane by regular hexagons. Consider the virus-containment process in whic…

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…