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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AIMS Press
2024-12-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241724 |
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| author | Kai An Sim Kok Bin Wong |
| author_facet | Kai An Sim Kok Bin Wong |
| author_sort | Kai An Sim |
| collection | DOAJ |
| description | 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)) $. |
| format | Article |
| id | doaj-art-16384d7b1fd94566a76832aee679f13c |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-16384d7b1fd94566a76832aee679f13c2025-01-23T07:53:26ZengAIMS PressAIMS Mathematics2473-69882024-12-01912363513637010.3934/math.20241724On the cooling number of the generalized Petersen graphsKai An Sim0Kok Bin Wong1School of Mathematical Sciences, Sunway University, 47500 Bandar Sunway, MalaysiaInstitute of Mathematical Sciences, Faculty of Science, Universiti Malaya, 50603 Kuala Lumpur, MalaysiaRecently, 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)) $.https://www.aimspress.com/article/doi/10.3934/math.20241724cooling numbergeneralized petersen graphs |
| spellingShingle | Kai An Sim Kok Bin Wong On the cooling number of the generalized Petersen graphs AIMS Mathematics cooling number generalized petersen graphs |
| title | On the cooling number of the generalized Petersen graphs |
| title_full | On the cooling number of the generalized Petersen graphs |
| title_fullStr | On the cooling number of the generalized Petersen graphs |
| title_full_unstemmed | On the cooling number of the generalized Petersen graphs |
| title_short | On the cooling number of the generalized Petersen graphs |
| title_sort | on the cooling number of the generalized petersen graphs |
| topic | cooling number generalized petersen graphs |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241724 |
| work_keys_str_mv | AT kaiansim onthecoolingnumberofthegeneralizedpetersengraphs AT kokbinwong onthecoolingnumberofthegeneralizedpetersengraphs |