On the cooling number of the generalized Petersen graphs

Recently, Bonato et al. (2024) introduced a new graph parameter, which is the cooling number of a graph $G$, denoted as CL $(G)$. In contrast with burning which seeks to minimize the number of rounds to burn all vertices in a graph, cooling seeks to maximize the number of rounds to cool all vertices in the graph. Cooling number is the compelling counterpart to the well-studied burning number, offering a new perspective on dynamic processes within graphs. In this paper, we showed that the cooling number of a classic cubic graph, the generalized Petersen graphs $P(n,k)$, is $\lfloor \frac{n}{2k}\rfloor +\lfloor \frac{k+1}{2}\rfloor +O(1)$ by the use of vertex-transitivity and combinatorial arguments. Particularly, we determined the exact values for CL $(P(n,1))$ and CL $(P(n,2))$.