On the cooling number of the generalized Petersen graphs

Abstract

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 $ \left\lfloor \frac{n}{2k} \right\rfloor + \left\lfloor \frac{k+1}{2} \right\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)) $.

Keywords

• cooling number
• generalized petersen graphs