Firefighting on the hexagonal grid

Abstract

The firefighter problem with k firefighters on an infinite graph G is an iterative graph process, defined as follows: Suppose a fire breaks out at a given vertex v∈V(G) on Turn 1. On each subsequent even turn, k firefighters protect k vertices that are not on fire, and on each subsequent odd turn, any vertex that is on fire spreads the fire to all adjacent unprotected vertices. The firefighters’ goal is to eventually stop the spread of the fire. If there exists a strategy for k firefighters to eventually stop the spread of the fire, then we say G is k-containable.We consider the firefighter problem on the hexagonal grid, which is the graph whose vertices and edges are exactly the vertices and edges of a regular hexagonal tiling of the plane. It is not known if the hexagonal grid is 1-containable. In Gavenčiak et al. (2014), it was shown that if the firefighters have one firefighter per turn and one extra firefighter on two turns, the firefighters can contain the fire. We improve on this result by showing that even with only one extra firefighter on one turn, the firefighters can still contain the fire.