Given a finite, simple, undirected graph , define cooling as a discrete-time process in which all nodes
start off uncooled. At each subsequent step, one new uncooled node (called a source)
is chosen to cool if such a node is available. If a node is cooled, then it remains
in that state until the end of the process. Once a node is cooled, its uncooled neighbors
become cooled in the next step. The process terminates when all nodes of are cooled, and the cooling number of is defined as the maximum number of steps for the cooling
process to end (Bonato et al. 2024).

The cooling number of a graph measures the speed of a slow-moving contagion in a graph such that the lower the cooling number, the faster the contagion spreads (Bonato
et al. 2024). Therefore, in contrast to the burning
number of a graph--which gives the minimum number of rounds to burn all nodes--the
cooling number gives the maximum number of rounds to cool all nodes.

where
is the burning number. This is because burning
number corresponds to the first iteration in which a completely burnt graph first
occurs, while the cooling number corresponds to the iteration in which no partially-uncooled
graph remains.