The detour covering number and cummerbund covering number of a graph
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916723682181120 |
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| author | Li, Chengli Zhan, Xingzhi |
| author_facet | Li, Chengli Zhan, Xingzhi |
| contents | We introduce several new concepts about graphs and investigate their basic properties. A longest path in a graph is called a detour and a longest cycle is called a cummerbund. The detour covering number of a graph is the number of vertices that lie in a detour. A graph is said to be detour covered if every vertex lies in a detour. The cummerbund covering number and cummerbund covered graphs are defined similarly. Some of the main results are as follows. (1) Minimum degree and forbidden subgraph conditions that ensure a graph to be cummerbund covered or detour covered. (2) The minimum cummerbund covering number and minimum detour covering number of a graph with connectivity or girth conditions. (3) The minimum cummerbund covering number of a $2$-connected bipartite graph and the extremal graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04131 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The detour covering number and cummerbund covering number of a graph Li, Chengli Zhan, Xingzhi Combinatorics 05C38, 05C35, 05C40, 05C07 We introduce several new concepts about graphs and investigate their basic properties. A longest path in a graph is called a detour and a longest cycle is called a cummerbund. The detour covering number of a graph is the number of vertices that lie in a detour. A graph is said to be detour covered if every vertex lies in a detour. The cummerbund covering number and cummerbund covered graphs are defined similarly. Some of the main results are as follows. (1) Minimum degree and forbidden subgraph conditions that ensure a graph to be cummerbund covered or detour covered. (2) The minimum cummerbund covering number and minimum detour covering number of a graph with connectivity or girth conditions. (3) The minimum cummerbund covering number of a $2$-connected bipartite graph and the extremal graphs. |
| title | The detour covering number and cummerbund covering number of a graph |
| topic | Combinatorics 05C38, 05C35, 05C40, 05C07 |
| url | https://arxiv.org/abs/2505.04131 |