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 verti...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2024-12-01
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| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241724 |
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| Summary: | 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)) $. |
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| ISSN: | 2473-6988 |